{"id":111952,"date":"2022-02-03T15:49:08","date_gmt":"2022-02-03T21:49:08","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=111952"},"modified":"2026-03-28T11:47:57","modified_gmt":"2026-03-28T16:47:57","slug":"nth-term-test-for-divergence","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/nth-term-test-for-divergence\/","title":{"rendered":"Nth Term Test for Divergence"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_g9fvd45BhKc\" 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_g9fvd45BhKc\" data-source-videoID=\"g9fvd45BhKc\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Nth Term Test for Divergence Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Nth Term Test for Divergence\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_g9fvd45BhKc:hover {cursor:pointer;} img#videoThumbnailImage_g9fvd45BhKc {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1728-thumb-final-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_g9fvd45BhKc\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_g9fvd45BhKc\" 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\",\"Nth Term Test for Divergence\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_g9fvd45BhKc\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_g9fvd45BhKc\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_g9fvd45BhKc\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction qig_Function() {\n  var x = document.getElementById(\"qig\");\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=\"qig_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=\"qig\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Sequence_vs_Series\" class=\"smooth-scroll\">Sequence vs. Series<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#Sequence\" class=\"smooth-scroll\">Sequence<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Series\" class=\"smooth-scroll\">Series<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"toc-h2\"><a href=\"#Nth_Term_Test\" class=\"smooth-scroll\">Nth Term Test<\/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>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Today we\u2019re going to work with <strong>limits<\/strong>, specifically the limit of a series. In order to learn something about the limit of a series, we can run a test on that sequence that creates the series.<\/p>\n<h2><span id=\"Sequence_vs_Series\" class=\"m-toc-anchor\"><\/span>Sequence vs. Series<\/h2>\n<p>\nBut before we get into that, let\u2019s back up a bit and get our vocabulary straight by reviewing how a sequence and a series are different. <\/p>\n<h3><span id=\"Sequence\" class=\"m-toc-anchor\"><\/span>Sequence<\/h3>\n<p>\nA sequence is a list of numbers in a particular order, like this: <\/p>\n<div class=\"examplesentence\">\\({1,3,5,7,9,11,13,&#8230;}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can also write a sequence as a formula or rule. For our sequence it looks like this:<\/p>\n<div class=\"examplesentence\">\\(a_{n}=2n-1\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe letter \\(n\\) represents the number of the term in the sequence. So the second number in the sequence can be found by substituting 2 for \\(n\\) in the formula. <\/p>\n<div class=\"examplesentence\">\\(a_{2}=2(2)-1=3\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Series\" class=\"m-toc-anchor\"><\/span>Series<\/h3>\n<p>\nA series is a bit different. A series is the sum of all the numbers in a sequence. So it looks like this: <\/p>\n<div class=\"examplesentence\">\\({1+3+5+7+9+11+13+&#8230;}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNotice that instead of commas we have addition signs. Each term also has a partial sum: <\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 80%;\">\n<tbody>\n<tr>\n<td><strong>Term Number<\/strong><\/td>\n<td>1    2    3    4    5     6   7    8<\/td>\n<\/tr>\n<tr>\n<td><strong>Series<\/strong><\/td>\n<td>1 + 3 + 5 + 7 + 9 +11 + 13 +15<\/td>\n<\/tr>\n<tr>\n<td><strong>Partial Sum<\/strong><\/td>\n<td>1   4    9    16    25    36    49    64<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThe <strong>partial sum<\/strong> is the sum of all the numbers in the series up until that point. So the partial sum of the third term is \\(1+3+5\\), which is 9. And if we look carefully at our table, we can see <br \/>\nsomething very interesting if we look for the patterns.<\/p>\n<p>Our series is adding the odd positive integers, and the partial sums also look very familiar. They are the perfect squares!<\/p>\n<p>We can also write a series using math notation:<\/p>\n<div class=\"examplesentence\">\\(\\displaystyle \\sum_{n=1}^{\\infty }2n-11\\)<\/div>\n<p>\n&nbsp;<br \/>\nLook at that fancy symbol! That\u2019s our summation symbol, the Greek letter sigma. And it just means to \u201csum up\u201d in math.<\/p>\n<p>So this expression is asking for the sum of the values of the expression \\(2n-1\\) for all terms from the first one until infinity. But for our purposes when we see the summation symbol sigma, we know that we are dealing with a <em>series<\/em>.<\/p>\n<p>When we see the infinity symbol above the sigma, we know that we are dealing with an <strong>infinite series<\/strong>. In fact, if we see the sigma without anything above or below it, we can assume that it is an infinite series of the terms from 1 to \\(\\infty\\). <\/p>\n<h2><span id=\"Nth_Term_Test\" class=\"m-toc-anchor\"><\/span>Nth Term Test<\/h2>\n<p>\nIn both science and finance, an infinite series that converges on an actual number is especially useful, so determining whether a series is divergent or convergent is important. In order for a series to be considered convergent, it must pass a sequence of tests. The first test that is used is the \\(n^{\\text{th}}\\) <strong>term test<\/strong>.<\/p>\n<p>The \\(n^{\\text{th}}\\) term test can tell us quickly if a series is divergent. It does not tell us if a series is convergent. If a series \u201cpasses\u201d the \\(n^{\\text{th}}\\) term test, then it must go through a bunch of other tests to be considered convergent. But the \\(n^{\\text{th}}\\) term test is very useful to rule out a divergent series so that all those other tests don\u2019t need to be completed. <\/p>\n<p>So what does the \\(n^{\\text{th}}\\) term test state? Here it is: <\/p>\n<div class=\"examplesentence\">When \\(\\displaystyle \\lim_{ n\\to \\infty}a_{n}\\neq 0\\) then \\(\\displaystyle \\sum_{n=1}^{\\infty }a_{n}\\) is divergent.<\/div>\n<p>\n&nbsp; <br \/>\nThis looks complicated but means that when the sequence \\(a_{n}\\) does not converge on 0, the series won\u2019t converge on anything. In other words, the series is definitely divergent. <\/p>\n<p>Let\u2019s go back to our series of odd numbers that sums to perfect squares and run this test. <\/p>\n<div class=\"examplesentence\">When \\(\\displaystyle \\sum_{n=1}^{\\infty }2n-1\\)<\/div>\n<p>\n&nbsp; <br \/>\nAccording to the \\(n^{\\text{th}}\\) term test, we need to determine if the sequence with the formula \\(2n-1\\) converges on 0 as \\(n\\rightarrow \\infty\\).<\/p>\n<p>The first few terms of our series look like this: <\/p>\n<div class=\"examplesentence\">\\({1+3+5+7+9+11+13+&#8230;}\\)<\/div>\n<p>\n&nbsp; <br \/>\nThe sequence of that rule looks the same but with commas used instead of addition symbols:<\/p>\n<div class=\"examplesentence\">\\({1,3,5,7,9,11,13,&#8230;}\\)<\/div>\n<p>\n&nbsp; <br \/>\nWe can easily see that this sequence is not approaching 0. In fact, it\u2019s approaching infinity, as the number will continue to increase by 2 forever. So this sequence is divergent. <\/p>\n<p>That means our series is also divergent because the sequence did not converge on 0. <\/p>\n<p>Already we can see that this test is going to rule out most series that are tested. Let\u2019s test a series that might have a chance.<\/p>\n<div class=\"examplesentence\">\\(\\displaystyle \\sum_{n=1}^{\\infty }\\frac{1}{n}\\)<\/div>\n<p>\n&nbsp; <br \/>\nLet\u2019s apply the \\(n^{\\text{th}}\\) term test to this series. Let\u2019s quickly review what the \\(n^{\\text{th}}\\) term test states:<\/p>\n<div class=\"examplesentence\">When \\(\\displaystyle \\lim_{ n\\to 0}a_{n}\\neq 0\\) , then \\(\\displaystyle \\sum_{n=1}^{\\infty }a_{n}\\) is divergent.<\/div>\n<p>\n&nbsp; <br \/>\nHere our \\(a_{n}\\) is \\(\\frac{1}{n}\\). So we need to find out if it converges on 0 or not: <\/p>\n<div class=\"examplesentence\">Does \\(\\displaystyle \\lim_{ n\\to \\infty }\\frac{1}{n}=0\\)?<\/div>\n<p>\n&nbsp; <br \/>\nIf we remember how limits of sequences work, we\u2019ll quickly recall that \\(\\frac{1}{n}\\) is one of our foundational sequences and does in fact converge on 0. Let\u2019s look at the first several terms: <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{1}{1},\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\frac{1}{6},\\frac{1}{7},\\frac{1}{8}\\)<\/div>\n<p>\n&nbsp; <br \/>\nWe can see that the denominator will keep getting bigger and bigger, making our fraction move closer and closer to 0 without ever getting there. <\/p>\n<p>So our series passes the \\(n^{\\text{th}}\\) term test! So what does this mean? It means that the infinite series \\(\\frac{1}{n}\\) may or may not be convergent. That\u2019s all the test tells us.<\/p>\n<p>To find out if it really is one of those special convergent series, we\u2019d have to run more mathematical tests. Spoiler alert\u2026it isn\u2019t a convergent series. In case you are curious, here are a couple that are convergent:<\/p>\n<div class=\"examplesentence\">\\(\\displaystyle \\sum_{n=1}^{\\infty }\\frac{1}{n^{2}}\\) and \\(\\sum_{n=1}^{\\infty }\\frac{1}{2^{n-1}}\\)<\/div>\n<p>\n&nbsp; <br \/>\nCan you see why they each passes the \\(n^{\\text{th}}\\) term test and are worthy of further consideration? (note: their sequences both converge on 0) <\/p>\n<p>That\u2019s it for the \\(n^{\\text{th}}\\) term test for determining whether a series is divergent. I hope this video was helpful. Thanks for watching, and happy studying!<\/p>\n<ul class=\"citelist\">\n<li><a href=\"https:\/\/www.mathsisfun.com\/algebra\/infinite-series.html\"target=\"_blank\">\u201cInfinite Series.\u201d n.d. <\/a><\/li>\n<li><a href=\"https:\/\/www.storyofmathematics.com\/nth-term-test\"target=\"_blank\">\u201cNth Term Test &#8211; Conditions, Explanation, and Examples.\u201d n.d. <\/a><\/li>\n<li><a href=\"https:\/\/www.mathsisfun.com\/algebra\/infinite-series.html\"target=\"_blank\">\u201cInfinite Series.\u201d n.d.<\/a><\/li>\n<li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Convergent_series\"target=\"_blank\">\u201cConvergent Series.\u201d Wikipedia<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":22,"featured_media":111955,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-111952","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/111952","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\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=111952"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/111952\/revisions"}],"predecessor-version":[{"id":281096,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/111952\/revisions\/281096"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/111955"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=111952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}