{"id":63850,"date":"2020-12-10T19:01:21","date_gmt":"2020-12-10T19:01:21","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=63850"},"modified":"2026-03-26T10:00:26","modified_gmt":"2026-03-26T15:00:26","slug":"infinite-series","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/infinite-series\/","title":{"rendered":"Infinite Series"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_j8PLimI2h6c\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_j8PLimI2h6c\" data-source-videoID=\"j8PLimI2h6c\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Infinite Series Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Infinite Series\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_j8PLimI2h6c:hover {cursor:pointer;} img#videoThumbnailImage_j8PLimI2h6c {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1727-infinite-series-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_j8PLimI2h6c\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_j8PLimI2h6c\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Infinite Series\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_j8PLimI2h6c\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_j8PLimI2h6c\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_j8PLimI2h6c\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction tw9_Function() {\n  var x = document.getElementById(\"tw9\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"tw9_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"tw9\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_an_Infinite_Series\" class=\"smooth-scroll\">What is an Infinite Series?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Sigma_Notation\" class=\"smooth-scroll\">Sigma Notation<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Convergence_vs_Divergence\" class=\"smooth-scroll\">Convergence vs. Divergence<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Practice_Questions\" class=\"smooth-scroll\">Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about infinite series!<\/p>\n<p>In this video, we will explore what an infinite series is, convergence vs. divergence, properties of convergent series, and the nth term test for divergence.<\/p>\n<p>Let\u2019s get started!<\/p>\n<h2><span id=\"What_is_an_Infinite_Series\" class=\"m-toc-anchor\"><\/span>What is an Infinite Series?<\/h2>\n<p>\nSo first, what is an infinite series? An infinite series is an infinite sum, which looks like this:<\/p>\n<div class=\"examplesentence\"><span style=\"font-size:90%\">\\(a_{1} + a_{2} + a_{3} + a_{4} + \u2026 + a_{n} + \u2026\\)<\/span><\/div>\n<p>\n&nbsp;<br \/>\nThe a\u2019s in the expression are called <strong>terms<\/strong> and can be pretty much anything\u2014numbers, functions, etc.\u2014and the subscripts represent the term number, kind of like a counter. In this series, \\(a_1\\) is the first term of the series, \\(a_2\\) is the second term, and so on.<\/p>\n<h2><span id=\"Sigma_Notation\" class=\"m-toc-anchor\"><\/span>Sigma Notation<\/h2>\n<p>\nWhen writing out equations, sigma notation is typically used to represent infinite sums. Instead of the expression we just saw, we can use a capital Greek sigma to write \\(\\sum_{n=1}^{\\infty}a_{n}\\) or simply \\(\\sum a_n\\) when the series is infinite.<\/p>\n<p>Suppose we had the series \\(\\sum_{n=1}^{\\infty}2n+1\\). This is what the first five terms written out would look like:<\/p>\n<ul>\n<li>When \\(n = 1\\): \\(2(1) + 1 = 3\\)<\/li>\n<li>When \\(n = 2\\): \\(2(2) + 1 = 5\\)<\/li>\n<li>When \\(n = 3\\): \\(2(3) + 1 = 7\\)<\/li>\n<li>When \\(n = 4\\): \\(2(4) + 1 = 9\\)<\/li>\n<li>When \\(n = 5\\): \\(2(5) + 1 = 11\\)<\/li>\n<\/ul>\n<p>So the first five terms of the series are \\(3+5+7+9+11\\).<\/p>\n<p>Usually, with infinite series, we are not interested in just a few terms\u2014we are interested in all of them.<\/p>\n<h2><span id=\"Convergence_vs_Divergence\" class=\"m-toc-anchor\"><\/span>Convergence vs. Divergence<\/h2>\n<p>\nNow, let\u2019s talk about convergence and divergence. If we were able to add up all of the terms in an infinite series, we would find that the addition of more and more terms would either \u201csettle\u201d on a value or keep growing.<\/p>\n<p>The series \\(\\sum_{n=1}^{\\infty} \\frac{1}{3n &#8211; 1}\\) equals \\(\\frac{1}{1}\\), \\(\\frac{1}{3}\\), \\(\\frac{1}{9}\\), \\(\\frac{1}{27}\\), and so on.<\/p>\n<p>In other words, the series is convergent and it converges on the value \\(\\frac{3}{2}\\). Notice how, when \\(n\\) gets large enough, adding \\(\\frac{1}{3^{n-1}}\\) won\u2019t really affect the sum because each successive fraction will be smaller than the one before.<\/p>\n<p>On the other hand, our first example series, \\(\\sum_{n=1}^{\\infty}2n+1 = 3 + 5 + 7 + 9 + 11\\), just keeps growing and is <strong>divergent<\/strong>. Each term is larger than the last, causing the sum to keep getting larger and diverge.<\/p>\n<p>To put it simply, convergent series have a limit; divergent series do not.<\/p>\n<h3><span id=\"Convergent_Series\" class=\"m-toc-anchor\"><\/span>Convergent Series<\/h3>\n<p>\nOnce you have determined that two series are convergent, there are three important properties to remember:<\/p>\n<ol>\n<li style=\"margin-bottom: 1em;\">The series of a sum is the sum of the series:<br \/>\n\\(\\sum(a_{n}+b_{n})=\\sum a_{n}+\\sum b_{n}\\)<\/li>\n<li style=\"margin-bottom: 1em;\">The series of a difference is the difference of the series:<br \/>\n\\(\\sum(a_{n}-b_{n})=\\sum a_{n}-\\sum b_{n}\\)<\/li>\n<li>The series of a multiple is the multiple of the series:<br \/>\n\\(\\sum ka_{n}=k\\sum a_{n}\\)<\/li>\n<\/ol>\n<p>Here are two convergent series to illustrate:<\/p>\n<ol>\n<li style=\"margin-bottom: 1em;\">The series of a sum is the sum of the series:<br \/>\n\\(\\sum \\left( \\frac{1}{n^2} + \\frac{1}{n^n} \\right) \\)\\(= \\sum \\frac{1}{n^2} + \\sum \\frac{1}{n^n} \\)\\(\\approx \\frac{\\pi^2}{6} + 1.29 \\)\\(\\approx 2.93<br \/>\n\\)<\/li>\n<li style=\"margin-bottom: 1em;\">The series of a difference is the difference of the series:<br \/>\n\\(\\sum \\left( \\frac{1}{n^2} &#8211; \\frac{1}{n^n} \\right) \\)\\(= \\sum \\frac{1}{n^2} &#8211; \\sum \\frac{1}{n^n} \\)\\(\\approx \\frac{\\pi^2}{6} &#8211; 1.29 \\approx 0.35<br \/>\n\\)<\/li>\n<li>The series of a multiple is the multiple of the series:<br \/>\n\\(\\sum \\left( 2 \\times \\frac{1}{n^n} \\right) \\)\\(= 2 \\sum \\frac{1}{n^n} \\)\\(\\approx 2 \\times 1.29 \\approx 2.58<br \/>\n\\)<\/li>\n<\/ol>\n<h3><span id=\"Divergent_Series\" class=\"m-toc-anchor\"><\/span>Divergent Series<\/h3>\n<p>\nNow let\u2019s shift to divergence. One of the first tests to perform on an infinite series is the \\(n\\)th term test for divergence. The advantage of this test is that if a series is divergent, you can stop there. The disadvantage is that knowing a series is not divergent by this test is not enough to automatically call it convergent, so the test becomes inconclusive.<\/p>\n<p>How does this test work? Recall, we said that a series is convergent if it has a limit and divergent if it does not. All we need to do is take the limit of the \\(n\\)th term of the sequence. If the limit is not equal to 0, the series is divergent. If the limit is equal to 0, our result is inconclusive.<\/p>\n<p>Back to our first example. We know that \\(\\sum_{n=1}^{\\infty} 2n=1\\) is divergent. The test confirms this.<\/p>\n<p>Since this limit is not equal to 0, the series is divergent.<\/p>\n<p>We know that \\(\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}\\) is convergent, but this test alone doesn\u2019t give us much to go on.<\/p>\n<p>This is inconclusive because the limit is equal to zero. As mentioned before, there are other more involved tests to determine whether series converge or diverge, but we\u2019ll cover that in another video.<\/p>\n<p>I hope this video helped you understand infinite series and how they work! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\">Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following series is considered a convergent series? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(1+3+5+7+9+&#8230;\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">\\(1+\\frac{1}{2}+\\frac{1}{4}\\)\\(+\\frac{1}{8}+\\frac{1}{16}+\\frac{1}{32}+&#8230;\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(\\frac{1}{8}+\\frac{1}{4}+\\frac{1}{2}\\)\\(+1+2+4+8+&#8230;\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(1+1-1+1-1+1&#8230;\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A convergent series is a series whose partial sums gets closer and closer to settling on a specific number, also called a limit. A divergent series does not approach a limit.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat will the third term be in the series below?<\/p>\n<div class=\"yellow-math-quote\">\\(\\sum_{n=1}^{\\infty}5n-2\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">11<\/div><div class=\"PQ\"  id=\"PQ-2-2\">12<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">13<\/div><div class=\"PQ\"  id=\"PQ-2-4\">14<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Substituting 3 for \\(n\\) results in \\(5(3)-2\\), which simplifies to 13.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nAn infinite series is _______________.<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">an infinite sum<\/div><div class=\"PQ\"  id=\"PQ-3-2\">an infinite limit<\/div><div class=\"PQ\"  id=\"PQ-3-3\">a divergent limit<\/div><div class=\"PQ\"  id=\"PQ-3-4\">a convergent limit<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>An infinite series is an infinite number of terms added together.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA series is _____________ if it has a limit, and ___________ if it does not have a limit. <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">divergent; convergent<\/div><div class=\"PQ\"  id=\"PQ-4-2\">infinite; divergent<\/div><div class=\"PQ\"  id=\"PQ-4-3\">convergent; infinite<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">convergent; divergent<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When the sum of a series approaches a limit it is considered convergent. If the series does not get closer and closer to a limit, it is considered divergent.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nIs the following series convergent or divergent?<\/p>\n<div class=\"yellow-math-quote\">\\(\\sum_{n=1}^{\\infty}4n+1\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Convergent<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-2\">Divergent<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The nth-term test for divergence can be performed on this series to determine if it is divergent or convergent. A series is convergent if it has a limit, and divergent if it does not have a limit. <\/p>\n<p>Take the limit of the nth-term of the sequence, and if the limit is not equal to zero, the series is divergent. If the limit is equal to zero, the result is inconclusive. In this example, the limit of the the nth-term of the sequence is not equal to zero, therefore it is divergent.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":100723,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-63850","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_type-video","8":"content_type-practice-questions","9":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/63850","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=63850"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/63850\/revisions"}],"predecessor-version":[{"id":250864,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/63850\/revisions\/250864"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100723"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=63850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}