<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@d6691e5af40348168a78e7f33c04634d" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Introduction</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@f875db9e06f74a4ea0f943c1d2a20f5a">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@f875db9e06f74a4ea0f943c1d2a20f5a" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Read <a href="/assets/courseware/v1/4fb79a8b271d75485a489411764ef691/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session23.pdf" target="[object Object]">Chapter 13.1–13.5 (PDF)</a> of <em>Mathematics for Computer Science</em> for 3.1 Sums & Products.</p>
<p>View the <a href="/assets/courseware/v1/eabc76e9ecd3f7eda8d0e9dd0d2a4dc4/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp23.pdf" target="[object Object]">Section 3.1 In-Class Questions (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@1a8e3a67809e42cc98f455ebe38e7611" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Lecture Video | Arithmetic Sums</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="video" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Arithmetic Sums</h3>
<div
id="video_7a37738852fb4dd4bba3aba8b84d29f3"
class="video closed"
data-metadata='{"savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "speed": null, "start": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/transcript/available_translations", "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/publish_completion", "completionPercentage": 0.95, "generalSpeed": 1.0, "completionEnabled": false, "prioritizeHls": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/transcript/translation/__lang__", "captionDataDir": null, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_arithmeticsum_video_ipod.mp4"], "poster": null, "ytMetadataEndpoint": "", "showCaptions": "true", "end": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/xmodule_handler/save_user_state", "autoplay": false, "transcriptLanguage": "en", "duration": 0.0, "transcriptLanguages": {"en": "English"}, "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "ytTestTimeout": 1500, "streams": "1.00:v6axtBS6IF8"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="7a37738852fb4dd4bba3aba8b84d29f3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_7a37738852fb4dd4bba3aba8b84d29f3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_7a37738852fb4dd4bba3aba8b84d29f3">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_arithmeticsum_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@7a37738852fb4dd4bba3aba8b84d29f3/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@528538da836e4d5a9709e7f6343abefe">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@528538da836e4d5a9709e7f6343abefe" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/d137c1c15a7c4380c11688e3140ecc5c/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_Arithmetic.pdf" target="[object Object]">Arithmetic Sums (PDF)</a></p>
<p><a href="/assets/courseware/v1/0c113aa813a04da4d2bc143141603c42/asset-v1:OCW+6.042J+2T2019+type@asset+block/ArithmeticSums_3.1_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@75afcc27a9ef4229ba207d40ed3f01d6" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Perturbation by Young Gauss</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_27c38f5e93644d709914b1338db5d1c5" class="problems-wrapper" role="group"
aria-labelledby="27c38f5e93644d709914b1338db5d1c5-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="27c38f5e93644d709914b1338db5d1c5-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5-problem-progress" tabindex="-1">
Perturbation by Young Gauss
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@27c38f5e93644d709914b1338db5d1c5-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>What perturbation did the nine-year-old C.F. Gauss reportly use to quickly sum 30 integers in increments of 13 starting with 89?</p>
<div class="inputtype option-input ">
<select name="input_27c38f5e93644d709914b1338db5d1c5_2_1" id="input_27c38f5e93644d709914b1338db5d1c5_2_1" aria-describedby="status_27c38f5e93644d709914b1338db5d1c5_2_1">
<option value="option_27c38f5e93644d709914b1338db5d1c5_2_1_dummy_default">Select an option</option>
<option value="find the average of the 30 integers and then multiply it by 30"> find the average of the 30 integers and then multiply it by 30</option>
<option value="subtract a shifted sum from the initial sum to get a massive cancellation of terms"> subtract a shifted sum from the initial sum to get a massive cancellation of terms</option>
<option value="add the sum to itself with the terms in reverse order to get 30 * (89 + 479)"> add the sum to itself with the terms in reverse order to get 30 * (89 + 479)</option>
<option value="apply the closed-form expression for arithmetic sums"> apply the closed-form expression for arithmetic sums</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_27c38f5e93644d709914b1338db5d1c5_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_27c38f5e93644d709914b1338db5d1c5_2_1"/>
</div></div>
<div class="solution-span">
<span id="solution_27c38f5e93644d709914b1338db5d1c5_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Perturbation by Young Gauss" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_27c38f5e93644d709914b1338db5d1c5" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_27c38f5e93644d709914b1338db5d1c5">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="27c38f5e93644d709914b1338db5d1c5-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27c38f5e93644d709914b1338db5d1c5-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27c38f5e93644d709914b1338db5d1c5-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27c38f5e93644d709914b1338db5d1c5-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@2a3011f06a2849f2bbc3013e1cea84a8" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Lecture Video | Geometric Sums</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="video" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Geometric Sums</h3>
<div
id="video_ab05810dd8354241aa51109dea5635ab"
class="video closed"
data-metadata='{"savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "speed": null, "start": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/transcript/available_translations", "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/publish_completion", "completionPercentage": 0.95, "generalSpeed": 1.0, "completionEnabled": false, "prioritizeHls": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/transcript/translation/__lang__", "captionDataDir": null, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_geometricsum_video_ipod.mp4"], "poster": null, "ytMetadataEndpoint": "", "showCaptions": "true", "end": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/xmodule_handler/save_user_state", "autoplay": false, "transcriptLanguage": "en", "duration": 0.0, "transcriptLanguages": {"en": "English"}, "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "ytTestTimeout": 1500, "streams": "1.00:ZDQk45NQbEo"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="ab05810dd8354241aa51109dea5635ab"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_ab05810dd8354241aa51109dea5635ab">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_ab05810dd8354241aa51109dea5635ab">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_geometricsum_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@ab05810dd8354241aa51109dea5635ab/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@26800cff1b23428c936046e25b9cde12">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@26800cff1b23428c936046e25b9cde12" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/c6c05f8c32740c28ba7eaa8a6eec7f0b/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_GeometricSum.pdf" target="[object Object]">Geometric Sums (PDF)</a></p>
<p><a href="/assets/courseware/v1/fc1536e315d6689fa37bd8d5b8912cbf/asset-v1:OCW+6.042J+2T2019+type@asset+block/GeometricSums_3.1_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@81403d1b81a942419f64b88260e5a9b6" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Annuities</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_3d6cd6a6500d4f4ca06944412d737d17" class="problems-wrapper" role="group"
aria-labelledby="3d6cd6a6500d4f4ca06944412d737d17-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="3d6cd6a6500d4f4ca06944412d737d17-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17-problem-progress" tabindex="-1">
Annuities
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@3d6cd6a6500d4f4ca06944412d737d17-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Assuming you could be sure of getting a 4% return forever on the money you invest, what would be a proper amount to invest now for an annuity that would pay you $10,000 yearly forever starting a year from now?</p>
<div id="inputtype_3d6cd6a6500d4f4ca06944412d737d17_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_3d6cd6a6500d4f4ca06944412d737d17_2_1" id="label_3d6cd6a6500d4f4ca06944412d737d17_2_1"><em>Enter the whole dollar amount without commas or the dollar sign. For example, if you want to invest $140,000.00, you should type 140000.</em></label>
<input type="text" name="input_3d6cd6a6500d4f4ca06944412d737d17_2_1" id="input_3d6cd6a6500d4f4ca06944412d737d17_2_1" aria-describedby="status_3d6cd6a6500d4f4ca06944412d737d17_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_3d6cd6a6500d4f4ca06944412d737d17_2_1"/>
<span class="status unanswered" id="status_3d6cd6a6500d4f4ca06944412d737d17_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_3d6cd6a6500d4f4ca06944412d737d17_2_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_3d6cd6a6500d4f4ca06944412d737d17_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Annuities" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_3d6cd6a6500d4f4ca06944412d737d17" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_3d6cd6a6500d4f4ca06944412d737d17">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="3d6cd6a6500d4f4ca06944412d737d17-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="3d6cd6a6500d4f4ca06944412d737d17-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="3d6cd6a6500d4f4ca06944412d737d17-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="3d6cd6a6500d4f4ca06944412d737d17-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@cc34762b50cd4864a9cea6903e777327" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Lecture Video | Book Stacking</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="video" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Book Stacking </h3>
<div
id="video_17a6c379c79140b0a8e9749c9e281ecd"
class="video closed"
data-metadata='{"savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "speed": null, "start": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/transcript/available_translations", "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/publish_completion", "completionPercentage": 0.95, "generalSpeed": 1.0, "completionEnabled": false, "prioritizeHls": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/transcript/translation/__lang__", "captionDataDir": null, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_bookstack_video_ipod.mp4"], "poster": null, "ytMetadataEndpoint": "", "showCaptions": "true", "end": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/xmodule_handler/save_user_state", "autoplay": false, "transcriptLanguage": "en", "duration": 0.0, "transcriptLanguages": {"en": "English"}, "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "ytTestTimeout": 1500, "streams": "1.00:CdhuVhWTSMI"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="17a6c379c79140b0a8e9749c9e281ecd"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_17a6c379c79140b0a8e9749c9e281ecd">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_17a6c379c79140b0a8e9749c9e281ecd">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_bookstack_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@17a6c379c79140b0a8e9749c9e281ecd/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@743070b790c74aee85ce656ec9ddea61">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@743070b790c74aee85ce656ec9ddea61" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/9e529872b40faaff16e5ebdeb49cf04f/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_BookStacking.pdf" target="[object Object]">Book Stacking (PDF)</a></p>
<p><a href="/assets/courseware/v1/ccaf5c30d5bcc832f4e0ae4a3f5e98ab/asset-v1:OCW+6.042J+2T2019+type@asset+block/BookStacking_3.1_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@4705e26f3e8748bb857105e170fbd797" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Harmonic Numbers</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_44d017f7b0a2442bb6d97d4facb4eb83" class="problems-wrapper" role="group"
aria-labelledby="44d017f7b0a2442bb6d97d4facb4eb83-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="44d017f7b0a2442bb6d97d4facb4eb83-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83-problem-progress" tabindex="-1">
Harmonic Numbers
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@44d017f7b0a2442bb6d97d4facb4eb83-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Which of the following are true?</p>
<div class="choicegroup capa_inputtype" id="inputtype_44d017f7b0a2442bb6d97d4facb4eb83_2_1">
<fieldset aria-describedby="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1">
<div class="field">
<input type="checkbox" name="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1[]" id="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="44d017f7b0a2442bb6d97d4facb4eb83_2_1-choice_0-label" for="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1"> Each Harmonic number is defined in terms of a simple summation.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1[]" id="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="44d017f7b0a2442bb6d97d4facb4eb83_2_1-choice_1-label" for="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>H</mi>
<mi>n</mi>
</msub>
<mo>::=</mo>
<munderover>
<mo movablelimits="false">&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>n</mi>
</munderover>
<mfrac>
<mn>1</mn>
<mi>i</mi>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1[]" id="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="44d017f7b0a2442bb6d97d4facb4eb83_2_1-choice_2-label" for="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1"> The overhang of a stable stack of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> books on the edge of a table is equal to the <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math>-th Harmonic number.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1[]" id="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="44d017f7b0a2442bb6d97d4facb4eb83_2_1-choice_3-label" for="input_44d017f7b0a2442bb6d97d4facb4eb83_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1"> In calculating the <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#9651;<!-- &#9651; --></mo>
</math>-overhang for the <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math> -th book that ensures stable stacking, the expression <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#9651;<!-- &#9651; -->=</mo>
<mfrac>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math> comes from the equality <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mo>&#9651;<!-- &#9651; -->=</mo>
<mn>1</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mo stretchy="false">(</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>&#8722;<!-- &#8722; --></mo>
<mo>&#9651;<!-- &#9651; --></mo>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<span id="answer_44d017f7b0a2442bb6d97d4facb4eb83_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_44d017f7b0a2442bb6d97d4facb4eb83_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_44d017f7b0a2442bb6d97d4facb4eb83_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Harmonic Numbers" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_44d017f7b0a2442bb6d97d4facb4eb83" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_44d017f7b0a2442bb6d97d4facb4eb83">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="44d017f7b0a2442bb6d97d4facb4eb83-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="44d017f7b0a2442bb6d97d4facb4eb83-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="44d017f7b0a2442bb6d97d4facb4eb83-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="44d017f7b0a2442bb6d97d4facb4eb83-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@142a0ffa4a9940bfa3a2779191594772" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Lecture Video | Integral Method</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="video" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Integral Method</h3>
<div
id="video_bf1cefdea26f4614964ba748cd0886c3"
class="video closed"
data-metadata='{"savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "speed": null, "start": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/transcript/available_translations", "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/publish_completion", "completionPercentage": 0.95, "generalSpeed": 1.0, "completionEnabled": false, "prioritizeHls": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/transcript/translation/__lang__", "captionDataDir": null, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_integralsum_video_ipod.mp4"], "poster": null, "ytMetadataEndpoint": "", "showCaptions": "true", "end": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/xmodule_handler/save_user_state", "autoplay": false, "transcriptLanguage": "en", "duration": 0.0, "transcriptLanguages": {"en": "English"}, "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "ytTestTimeout": 1500, "streams": "1.00:EegG5TPL29c"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="bf1cefdea26f4614964ba748cd0886c3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_bf1cefdea26f4614964ba748cd0886c3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_bf1cefdea26f4614964ba748cd0886c3">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_integralsum_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@bf1cefdea26f4614964ba748cd0886c3/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@7f60fe99af8549da912823045432203d">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@7f60fe99af8549da912823045432203d" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/1cfe13da82ae78b8c1d65fd71333a79b/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_IntegralMeth.pdf" target="[object Object]">Integral Method (PDF)</a></p>
<p><a href="/assets/courseware/v1/977ff6b120fc23a8f2cf7c2ffa7b8092/asset-v1:OCW+6.042J+2T2019+type@asset+block/IntegralMethod_3.1_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@f57b318bc25c480db903f845ca22cf64" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Integral Method Demystified</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_94ea1513c4194b8e8e30fc37f0b8e60b" class="problems-wrapper" role="group"
aria-labelledby="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="8"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b-problem-progress" tabindex="-1">
Integral Method Demystified
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@94ea1513c4194b8e8e30fc37f0b8e60b-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_2_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-legend" class="response-fieldset-legend field-group-hd">Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">&#8594;<!-- &#8594; --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
<mo>::=</mo>
<munderover>
<mo movablelimits="false">&#8721;<!-- &#8721; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>::=</mo>
<munderover>
<mo>&#8747;<!-- &#8747; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>x</mi>
</math>.
<br/>
<br/>
1. What is the upper bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is weakly increasing?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi></math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-choice_3-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_2_1-choice_4-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_3_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-legend" class="response-fieldset-legend field-group-hd">2. What is the upper bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is weakly decreasing?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi></math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-choice_3-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_3_1-choice_4-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_4_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-legend" class="response-fieldset-legend field-group-hd">3. What is the lower bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is weakly increasing?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi></math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-choice_3-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_4_1-choice_4-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_5_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-legend" class="response-fieldset-legend field-group-hd">4. What is the lower bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is weakly decreasing?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi></math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-choice_3-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_5_1-choice_4-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>I</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_6_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_6_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_6_1-legend" class="response-fieldset-legend field-group-hd">5. Do the upper bounds and lower bounds for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> change if it is strictly increasing/decreasing instead of weakly increasing/decreasing?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_6_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_6_1"> yes
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_6_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_6_1"> no
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_6_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_6_1"> it depends
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_94ea1513c4194b8e8e30fc37f0b8e60b_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_7_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_7_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_7_1-legend" class="response-fieldset-legend field-group-hd">6. What is the upper bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>H</mi>
<mi>n</mi>
</msub>
</math>?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_7_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_7_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>1</mn>
<mo>+</mo>
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_7_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_7_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>?
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_7_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_7_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_7_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_7_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_8_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_8_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_8_1-legend" class="response-fieldset-legend field-group-hd">7. What is the lower bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>H</mi>
<mi>n</mi>
</msub>
</math>?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_8_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_8_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>1</mn>
<mo>+</mo>
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_8_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_8_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>?
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_8_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_8_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_8_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_8_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div class="choicegroup capa_inputtype" id="inputtype_94ea1513c4194b8e8e30fc37f0b8e60b_9_1">
<fieldset aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_9_1">
<legend id="94ea1513c4194b8e8e30fc37f0b8e60b_9_1-legend" class="response-fieldset-legend field-group-hd">8. What is the asymptotic bound for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>H</mi>
<mi>n</mi>
</msub>
</math>?</legend>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_9_1-choice_0-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_9_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>1</mn>
<mo>+</mo>
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_9_1-choice_1-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_9_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math>?
</label>
</div>
<div class="field">
<input type="radio" name="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1" id="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="94ea1513c4194b8e8e30fc37f0b8e60b_9_1-choice_2-label" for="input_94ea1513c4194b8e8e30fc37f0b8e60b_9_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_94ea1513c4194b8e8e30fc37f0b8e60b_9_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_94ea1513c4194b8e8e30fc37f0b8e60b_9_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_94ea1513c4194b8e8e30fc37f0b8e60b_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_94ea1513c4194b8e8e30fc37f0b8e60b_solution_2"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Integral Method Demystified " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_94ea1513c4194b8e8e30fc37f0b8e60b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_94ea1513c4194b8e8e30fc37f0b8e60b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="94ea1513c4194b8e8e30fc37f0b8e60b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@d779469bee3341ae87c9909845932a40" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Lecture Video | Stirling's Formula</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="video" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Stirling's Formula </h3>
<div
id="video_03591c779c1b4f50b2978a3d6b1bf688"
class="video closed"
data-metadata='{"savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "speed": null, "start": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/transcript/available_translations", "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/publish_completion", "completionPercentage": 0.95, "generalSpeed": 1.0, "completionEnabled": false, "prioritizeHls": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/transcript/translation/__lang__", "captionDataDir": null, "sources": ["https://ia600207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_stirling_video_ipod.mp4"], "poster": null, "ytMetadataEndpoint": "", "showCaptions": "true", "end": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/xmodule_handler/save_user_state", "autoplay": false, "transcriptLanguage": "en", "duration": 0.0, "transcriptLanguages": {"en": "English"}, "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "ytTestTimeout": 1500, "streams": "1.00:lU_QT5GSuxI"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="03591c779c1b4f50b2978a3d6b1bf688"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_03591c779c1b4f50b2978a3d6b1bf688">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_03591c779c1b4f50b2978a3d6b1bf688">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia600207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_stirling_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@03591c779c1b4f50b2978a3d6b1bf688/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@cc5a62db10304c77b89b7a13879e659e">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@cc5a62db10304c77b89b7a13879e659e" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/ec285edfb296b614e866241fc76a078b/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_StirlingForm.pdf" target="[object Object]">Stirling's Formula (PDF)</a></p>
<p><a href="/assets/courseware/v1/8b16ccbf8eeb8b60e3ad1cae7ecc77dd/asset-v1:OCW+6.042J+2T2019+type@asset+block/StirlingsFormula_3.1_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@ec4a00b5a8694848ba5ebc218b0e31ef" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Applying Sterling's Formula</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@9f14f2977215499d9603bf3470d2c28c">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@9f14f2977215499d9603bf3470d2c28c" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="html" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>The quantity</p>
<p></p>
<p></p>
<div align="center">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle mathsize="1.44em"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> </mfrac> <mspace width="2em"></mspace> <mspace width="2em"></mspace> </mstyle> </math></p>
</div>
<p>will come up later in the course (it is the probability that in <math xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </math> flips of a fair coin, exactly <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>n</mi> </math> will be Heads). Which of the following simplified formulas is asymptotically equal to the above formula?</p>
<p>(Hint: to find the final formula, write out a short sequence of successively simpler formulas.)</p>
<p></p>
<p></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_558fdf8d74e149dc8a9de74c2cfe66a0" class="problems-wrapper" role="group"
aria-labelledby="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0-problem-progress" tabindex="-1">
Question
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@558fdf8d74e149dc8a9de74c2cfe66a0-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_558fdf8d74e149dc8a9de74c2cfe66a0_2_1">
<fieldset aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1">
<div class="field">
<input type="radio" name="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" id="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="558fdf8d74e149dc8a9de74c2cfe66a0_2_1-choice_0-label" for="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<msqrt>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</msqrt>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" id="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="558fdf8d74e149dc8a9de74c2cfe66a0_2_1-choice_1-label" for="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</msqrt>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" id="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="558fdf8d74e149dc8a9de74c2cfe66a0_2_1-choice_2-label" for="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mfrac>
<mn>2</mn>
<mrow>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</mrow>
</mfrac>
</msqrt>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" id="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="558fdf8d74e149dc8a9de74c2cfe66a0_2_1-choice_3-label" for="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
<msqrt>
<mn>2</mn>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</msqrt>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" id="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="558fdf8d74e149dc8a9de74c2cfe66a0_2_1-choice_4-label" for="input_558fdf8d74e149dc8a9de74c2cfe66a0_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mn>2</mn>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</msqrt>
</math>
</label>
</div>
<span id="answer_558fdf8d74e149dc8a9de74c2cfe66a0_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_558fdf8d74e149dc8a9de74c2cfe66a0_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_558fdf8d74e149dc8a9de74c2cfe66a0_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Question" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_558fdf8d74e149dc8a9de74c2cfe66a0" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_558fdf8d74e149dc8a9de74c2cfe66a0">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="558fdf8d74e149dc8a9de74c2cfe66a0-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@66bd490514784ab88a9c7a7e0d5f0641" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Convergence of Geometric Series</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_481f23af96ff4e268ed5537d1d901b22" class="problems-wrapper" role="group"
aria-labelledby="481f23af96ff4e268ed5537d1d901b22-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="481f23af96ff4e268ed5537d1d901b22-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22-problem-progress" tabindex="-1">
Convergence of Geometric Series
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@481f23af96ff4e268ed5537d1d901b22-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_481f23af96ff4e268ed5537d1d901b22_2_1">
<fieldset aria-describedby="status_481f23af96ff4e268ed5537d1d901b22_2_1">
<legend id="481f23af96ff4e268ed5537d1d901b22_2_1-legend" class="response-fieldset-legend field-group-hd">For which of the following values of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
</math> does the sum of the geometric series with common ratio <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
</math> not converge?</legend>
<div class="field">
<input type="radio" name="input_481f23af96ff4e268ed5537d1d901b22_2_1" id="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="481f23af96ff4e268ed5537d1d901b22_2_1-choice_0-label" for="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_481f23af96ff4e268ed5537d1d901b22_2_1"> 0
</label>
</div>
<div class="field">
<input type="radio" name="input_481f23af96ff4e268ed5537d1d901b22_2_1" id="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="481f23af96ff4e268ed5537d1d901b22_2_1-choice_1-label" for="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_481f23af96ff4e268ed5537d1d901b22_2_1"> -0.5
</label>
</div>
<div class="field">
<input type="radio" name="input_481f23af96ff4e268ed5537d1d901b22_2_1" id="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="481f23af96ff4e268ed5537d1d901b22_2_1-choice_2-label" for="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_481f23af96ff4e268ed5537d1d901b22_2_1"> 0.5
</label>
</div>
<div class="field">
<input type="radio" name="input_481f23af96ff4e268ed5537d1d901b22_2_1" id="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="481f23af96ff4e268ed5537d1d901b22_2_1-choice_3-label" for="input_481f23af96ff4e268ed5537d1d901b22_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_481f23af96ff4e268ed5537d1d901b22_2_1"> 1
</label>
</div>
<span id="answer_481f23af96ff4e268ed5537d1d901b22_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_481f23af96ff4e268ed5537d1d901b22_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_481f23af96ff4e268ed5537d1d901b22_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Convergence of Geometric Series" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_481f23af96ff4e268ed5537d1d901b22" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_481f23af96ff4e268ed5537d1d901b22">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="481f23af96ff4e268ed5537d1d901b22-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="481f23af96ff4e268ed5537d1d901b22-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="481f23af96ff4e268ed5537d1d901b22-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="481f23af96ff4e268ed5537d1d901b22-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@f8c23d236bf744d0bc461cbc521457d5" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Summation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_d7868528278544f685d7594804637738" class="problems-wrapper" role="group"
aria-labelledby="d7868528278544f685d7594804637738-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="d7868528278544f685d7594804637738-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738-problem-progress" tabindex="-1">
Summation
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@d7868528278544f685d7594804637738-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_d7868528278544f685d7594804637738_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_d7868528278544f685d7594804637738_2_1" id="label_d7868528278544f685d7594804637738_2_1">There is a number <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> such that <math xmlns="http://www.w3.org/1998/Math/MathML">
<munderover>
<mo movablelimits="false">&#8721;<!-- &#8721; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</munderover>
<msup>
<mi>i</mi>
<mi>p</mi>
</msup>
</math> converges to a finite value iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
<mo>&lt;</mo>
<mi>a</mi>
</math>.
<br/>
<br/>
1. What is the value of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math>?</label>
<input type="text" name="input_d7868528278544f685d7594804637738_2_1" id="input_d7868528278544f685d7594804637738_2_1" aria-describedby="status_d7868528278544f685d7594804637738_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_d7868528278544f685d7594804637738_2_1"/>
<span class="status unanswered" id="status_d7868528278544f685d7594804637738_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_d7868528278544f685d7594804637738_2_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_d7868528278544f685d7594804637738_3_1">
<fieldset aria-describedby="status_d7868528278544f685d7594804637738_3_1">
<legend id="d7868528278544f685d7594804637738_3_1-legend" class="response-fieldset-legend field-group-hd">2. Which of the following would be good approaches as part of a proof that this value of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> is correct?</legend>
<div class="field">
<input type="checkbox" name="input_d7868528278544f685d7594804637738_3_1[]" id="input_d7868528278544f685d7594804637738_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="d7868528278544f685d7594804637738_3_1-choice_0-label" for="input_d7868528278544f685d7594804637738_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_d7868528278544f685d7594804637738_3_1"> find a closed form for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msubsup>
<mo>&#8747;<!-- &#8747; --></mo>
<mn>1</mn>
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</msubsup>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
<mspace width="mediummathspace"/>
<mi>d</mi>
<mi>x</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d7868528278544f685d7594804637738_3_1[]" id="input_d7868528278544f685d7594804637738_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="d7868528278544f685d7594804637738_3_1-choice_1-label" for="input_d7868528278544f685d7594804637738_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_d7868528278544f685d7594804637738_3_1"> find a closed form for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msubsup>
<mo>&#8747;<!-- &#8747; --></mo>
<mn>1</mn>
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</msubsup>
<msup>
<mi>i</mi>
<mi>x</mi>
</msup>
<mspace width="mediummathspace"/>
<mi>d</mi>
<mi>x</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d7868528278544f685d7594804637738_3_1[]" id="input_d7868528278544f685d7594804637738_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="d7868528278544f685d7594804637738_3_1-choice_2-label" for="input_d7868528278544f685d7594804637738_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_d7868528278544f685d7594804637738_3_1"> induction on <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d7868528278544f685d7594804637738_3_1[]" id="input_d7868528278544f685d7594804637738_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="d7868528278544f685d7594804637738_3_1-choice_3-label" for="input_d7868528278544f685d7594804637738_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_d7868528278544f685d7594804637738_3_1"> induction on <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> using the following sum: <math xmlns="http://www.w3.org/1998/Math/MathML">
<munderover>
<mo movablelimits="false">&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msup>
<mi>i</mi>
<mi>p</mi>
</msup>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d7868528278544f685d7594804637738_3_1[]" id="input_d7868528278544f685d7594804637738_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="d7868528278544f685d7594804637738_3_1-choice_4-label" for="input_d7868528278544f685d7594804637738_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_d7868528278544f685d7594804637738_3_1"> compare the series term-by-term with the Harmonic series
</label>
</div>
<span id="answer_d7868528278544f685d7594804637738_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_d7868528278544f685d7594804637738_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_d7868528278544f685d7594804637738_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Summation" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_d7868528278544f685d7594804637738" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_d7868528278544f685d7594804637738">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="d7868528278544f685d7594804637738-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d7868528278544f685d7594804637738-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d7868528278544f685d7594804637738-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d7868528278544f685d7594804637738-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@5d872919f72843948e25bc7416364a72" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | Sum's Upper Lower Bounds</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="True" data-block-type="problem" data-runtime-version="1" data-request-token="a3666e6cdbda11ef905012ab06d4b43d" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_2e267e1000754832a03928c963bbc9e8" class="problems-wrapper" role="group"
aria-labelledby="2e267e1000754832a03928c963bbc9e8-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="2e267e1000754832a03928c963bbc9e8-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8-problem-progress" tabindex="-1">
Sum&#39;s Upper Lower Bounds
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@2e267e1000754832a03928c963bbc9e8-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_2e267e1000754832a03928c963bbc9e8_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2e267e1000754832a03928c963bbc9e8_2_1" id="label_2e267e1000754832a03928c963bbc9e8_2_1">Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
<mtext>&#160;</mtext>
<mo>=</mo>
<mtext>&#160;</mtext>
<munderover>
<mo movablelimits="false">&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>57</mn>
</munderover>
<mfrac>
<mn>1</mn>
<mroot>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>7</mn>
</mrow>
<mn>3</mn>
</mroot>
</mfrac>
</math>. The Integral Method provides the lower and upper bounds on <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math>. By how much does the upper bound differ from the lower bound?</label>
<input type="text" name="input_2e267e1000754832a03928c963bbc9e8_2_1" id="input_2e267e1000754832a03928c963bbc9e8_2_1" aria-describedby="status_2e267e1000754832a03928c963bbc9e8_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2e267e1000754832a03928c963bbc9e8_2_1"/>
<span class="status unanswered" id="status_2e267e1000754832a03928c963bbc9e8_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2e267e1000754832a03928c963bbc9e8_2_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_2e267e1000754832a03928c963bbc9e8_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Sum&#39;s Upper Lower Bounds" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_2e267e1000754832a03928c963bbc9e8" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_2e267e1000754832a03928c963bbc9e8">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="2e267e1000754832a03928c963bbc9e8-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="2e267e1000754832a03928c963bbc9e8-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="2e267e1000754832a03928c963bbc9e8-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="2e267e1000754832a03928c963bbc9e8-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>