{"id":174,"date":"2019-08-27T17:10:45","date_gmt":"2019-08-27T17:10:45","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=174"},"modified":"2019-12-02T21:08:47","modified_gmt":"2019-12-02T21:08:47","slug":"infinite-series-convergence","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/infinite-series-convergence\/","title":{"rendered":"Infinite Series Convergence"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Convergence Tests for Infinite Series &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nIn this tutorial, we review some of the most common tests for the\nconvergence of an <b>infinite series<\/b>\n$$\n\\sum_{k=0}^{\\infty} a_k = a_0 + a_1 + a_2 + \\cdots\n$$\nThe proofs or these tests are interesting, so we urge you to look them\nup in your calculus text.\n\n<\/p>\n\n\n\n<p>\nLet\n\\begin{eqnarray*}\n\ts_0 &amp; = &amp; a_0 \\\\\n\ts_1 &amp; = &amp; a_1 \\\\\n\t&amp; \\vdots &amp; \\\\\n\ts_n &amp; = &amp; \\sum_{k=0}^{n} a_k \\\\\n\t&amp; \\vdots &amp; \n\\end{eqnarray*}\nIf the sequence $\\{ s_n \\}$ of <b>partial sums<\/b> converges to a limit\n$L$, then the series is said to <b>converge<\/b> to the <b>sum<\/b> $L$\nand we write\n\n<\/p>\n\n\n\n<p>\n\t\t$\\qquad$\n<\/p>\n\n\n\n<p>\n\t\t$$\n\t\t\\sum_{k=0}^{\\infty}a_k = L.\n\t\t$$\n<\/p>\n\n\n\n<p>\n\t\t$\\qquad\\qquad$\n<\/p>\n\n\n\n<p>\n\t\t\tFor $j \\ge 0$, $\\sum\\limits^{\\infty}_{k=0} a_k$ converges if and only\n\t\t\tif $\\sum\\limits_{k=j}^{\\infty} a_k$ converges, so in discussing\n\t\t\tconvergence we often just write $\\sum a_k$.\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nConsider the <b>geometric series<\/b>\n$$\n\\sum_{k=0}^{\\infty} x^k.\n$$\nThe $n^{th}$ partial sum is\n$$\ns_n = 1 + x + x^{2} + \\cdots + x^{n}.\n$$\nMultiplying both sides by $x$,\n$$\nxs_n = x + x^{2} + x^{3} + \\cdots + x^{n+1}.\n$$\nSubtracting the second equation from the first,\n$$\n(1-x)s_n = 1-x^{n+1},\n$$\nso for $x \\not= 1$,\n$$\ns_n = \\frac{1-x^{n+1}}{1-x}.\n$$\nFor $|x| &lt; 1$,\n$$\n\\lim_{n \\rightarrow \\infty} s_n = \\frac{1}{1-x}.\n$$\n\nIt is easy to see that $\\sum\\limits_{k=0}^{\\infty}x^{k}$ diverges for\n$|x| \\ge 1$.  Thus $\\sum\\limits_{k=0}^{\\infty}x^{k} = \\frac{1}{1-x}$\nfor $|x| &lt; 1$ and diverges for $|x| \\ge 1$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Divergence Test<\/h4>\n\n\n\n<p>\n\nIf $\\lim\\limits_{k \\rightarrow \\infty} a_k \\not= 0$, then\n$\\sum\\limits_{k=0}^{\\infty}a_k$ diverges.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nThe series $\\sum\\limits_{k=0}^{\\infty}\\frac{k}{2k+1}$ diverges, since\n$\\lim\\limits_{k \\rightarrow \\infty} \\frac{k}{2k+1} = 1\/2 \\not= 0$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Integral Test<\/h4>\n\n\n\n<p>\n\nLet $f(x)$ be continuous, decreasing, and positive for $x \\ge 1$.\nThen $\\sum\\limits_{k=1}^{\\infty}f(k)$ converges if and only if\n$\\int\\limits_{1}^{\\infty} f(x)dx$ converges.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\tConsider the <b>p-series<\/b>\n\t\t$$\n\t\t\\sum_{k=1}^{\\infty}\\frac{1}{k^p} = \\frac{1}{1^p} + \\frac{1}{2^p} +\n\t\t\\frac{1}{3^p} + \\cdots\n\t\t$$\n\t\tSince\n\n\t\t$$ \n\t\t\\int_{1}^{\\infty}\\frac{1}{x^{p}}dx = \\left\\{\\begin{array}{l@{,\\quad}l}\n\t\t\t\\left.\\frac{1}{1-p}x^{1-p}\\right|_{1}^{\\infty} &amp; p&gt;1 \\\\\n\t\t\t\\left.\\ln |x| \\right|_{1}^{\\infty} &amp; p=1 \\\\\n\t\t\t\\left.\\frac{1}{1-p}x^{1-p}\\right|_{1}^{\\infty} &amp; 0 &lt; p &lt; 1\n\t\t\\end{array}  \\right. = \\left\\{ \\begin{array}{c}\n\t\t\t\\frac{1}{1-p}\\\\\n\t\t\t\\infty \\\\\n\t\t\t\\infty,\n\t\t\\end{array}\\right.\n\t\t$$\n\t\tthe series converges for $p&gt;1$ and diverges for $0 &lt; p \\le 1$.\n<\/p>\n\n\n\n<p>\n\t\t$\\qquad$\n<\/p>\n\n\n\n<p>\n\t\t\tThe divergent p-series \n\t\t\t$$\n\t\t\t\\sum_{k=1}^{\\infty}\\frac{1}{k}\n\t\t\t$$\n\t\t\twith $p=1$ is called the <b>Harmonic Series.<\/b>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Comparison Test<\/h4>\n\n\n\n<p>\n\nLet $\\sum a_k$ and $\\sum b_k$ be series with non-negative terms.  If\n$a_k \\le b_k$ for all $k$ sufficiently large, then\n\n<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li> If $\\sum b_k$ converges, then $\\sum a_k$ also converges.\n\n\t<\/li><li> If $\\sum a_k$ diverges, then $\\sum b_k$ also diverges.\n\n<\/li><\/ol>\n\n\n\n<p>\n\nInformally, if the &#8220;larger&#8221; series converges, so does the\n&#8220;smaller.&#8221;  If the &#8220;smaller&#8221; series divers, so does the &#8220;larger.&#8221;\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Examples<\/h6>\n\n\n\n<li> Since $\\sum\\limits_{k=1}^{\\infty}\\frac{1}{k^2}$ converges, so\n\t\t\t\tdoes $\\sum\\limits_{k=1}^{\\infty}\\frac{1}{k^2 + 3}$.\n\t\t\t<\/li>\n\n\n\n<li> Since $\\sum\\limits_{k=1}^{\\infty}\\frac{1}{k}$ diverges, so does\n\t\t\t\t$\\sum\\limits_{k=1}^{\\infty}\\frac{1}{\\ln |k+1|}$.\n<p><\/p>\n<p align=\"center\">\n\t\t\t$\\qquad\\qquad$\n<\/p><p align=\"center\">\n<\/p><p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\t\t$\\frac{1}{k^2 +3} &lt; \\frac{1}{k^2}$ for all $k$.\n<\/p>\n<p align=\"center\">\n\t\t\t\t\t$\\frac{1}{\\ln |k+1|} &gt; \\frac{1}{k}$ for $k \\ge 2$.\n<\/p>\n<p><\/p>\n<p><\/p>\n\n<p>\n<\/p>\n<h4>Limit Comparison Test<\/h4>\n<p>\n\nLet $\\sum a_k$ and $\\sum b_k$ be series with positive terms.  If\n$$\n\\lim_{k \\rightarrow \\infty}\\frac{a_k}{b_k} = L\n$$\nwhere $0 &lt; L &lt; \\infty$ then $\\sum a_k$ and $\\sum b_k$ either both\nconverge or both diverge.\n\n<\/p>\n<h4>Example<\/h4>\n<p>\n\nThe series $\\sum\\limits_{k=1}^{\\infty}\\frac{k^2-1}{5k^3}$ diverges,\nsince $\\sum\\limits_{k=1}^{\\infty}\\frac{1}{k}$ diverges and\n$$\n\\lim_{k \\rightarrow \\infty} \\frac{\\frac{k^2-1}{5k^3}}{\\frac{1}{k}} =\n\\lim_{k \\rightarrow \\infty} \\frac{k^2-1}{5k^2} = \\frac{1}{5}.\n$$\n\n<\/p><p>\n<\/p><h4>Ratio Test<\/h4>\n<p>\n\nLet $\\sum a_k$ be a series with positive terms and suppose that\n$$\n\\lim_{k \\rightarrow \\infty}\\frac{a_{k+1}}{a_k} = L.\n$$\n\n<\/p><ol>\n\n\t<li> If $L &lt; 1$, then $\\sum a_k$ converges.\n\n\t<\/li><li> If $L &gt; 1$, then $\\sum a_k$ diverges.\n\n\t<\/li><li> If $L = 1$, then the test is inconclusive.\n\n<\/li><\/ol>\n\n<p>\n<\/p>\n<h4>Example<\/h4>\n<p>\n\nThe series $\\sum\\limits_{k=1}^{\\infty}\\frac{1}{k!}$ converges, since\n$$\n\\lim_{k \\rightarrow \\infty}\\frac{\\frac{1}{(k+1)!}}{\\frac{1}{k!}} =\n\\lim_{k \\rightarrow \\infty}\\frac{1}{k+1} = 0.\n$$\n\n<\/p>\n<h4>Root Test<\/h4>\n<p>\n\nLet $\\sum a_k$ be a series with non-negative terms and suppose that\n$$\n\\lim_{k \\rightarrow \\infty} (a_k)^{\\frac{1}{k}} = L.\n$$\n\n<\/p><ol>\n\n\t<li> If $L &lt; 1$, then $\\sum a_k$ converges.\n\n\t<\/li><li> If $L &gt; 1$, then $\\sum a_k$ diverges.\n\n\t<\/li><li> If $L = 1$, then the test is inconclusive.\n\n<\/li><\/ol>\n\n<p>\n<\/p>\n<h4>Example<\/h4>\n<p>\n\nThe series $\\sum\\limits_{k=0}^{\\infty} \\left( \\frac{k}{2k+1}\n\\right)^{k}$ converges, since \n$$\n\\lim_{k \\rightarrow \\infty}\n\\left[\\left(\\frac{k}{2k+1}\\right)^{k}\\right]^{\\frac{1}{k}} = \\lim_{k\n\\rightarrow \\infty} \\frac{k}{2k+1} = \\frac{1}{2}.\n$$\n\n<\/p><p>\n<\/p><h4>Alternating Series Test<\/h4>\n<p>\n\nConsider the <b>alternating series<\/b>\n$$\n\\sum_{k=0}^{\\infty}(-1)^{k}a_k\n$$\nwhere $a_k &gt; 0$ for all $k \\ge 0$.\n\n<\/p><p>\nIf $a_{k+1} &lt; a_k$ for all $k$ and $\\lim\\limits a_k = 0$, then\n$\\sum\\limits_{k=0}^{\\infty}(-1)^{k}a_k$ converges.\n\n<\/p>\n<h6>Example<\/h6>\n<p>\n\nThe series $\\sum\\limits_{k=0}^{\\infty} \\frac{(-1)^{k}}{k+1}$\nconverges, since $\\frac{1}{(k+1) + 1} &lt; \\frac{1}{k+1}$ and\n$\\lim\\limits_{k \\rightarrow \\infty}\\frac{1}{k+1} =0$.  This series is\n<b>conditionally convergent<\/b>, rather than <b>absolutely\nconvergent<\/b>, since\n$\\sum\\limits_{k=0}^{\\infty}\\left|\\frac{(-1)^{k}}{k+1}\\right| =\n\\sum\\limits_{k=0}^{\\infty}\\frac{1}{k+1}$ diverges.\n\n<\/p><hr color=\"blue\">\n\n<center>\n<h4>Key Concepts<\/h4>\n<\/center>\n\nThe infinite series\n$$\n\\sum_{k=0}^{\\infty}a_k\n$$\nconverges if the sequence of partial sums converges and diverges\notherwise.\n\n<p>\nFor a particular series, one or more of the common convergence tests\nmay be most convenient to apply.\n\n\n\n<!------------------------>\n\n\n<br>\n\n<\/p><hr>\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ3810\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p><p>\n\n\n\n<\/p>\n\n<p><\/p>\n\n\n\n<p><\/p>\n<\/li>\n","protected":false},"excerpt":{"rendered":"<p>Convergence Tests for Infinite Series &#8211; HMC Calculus Tutorial In this tutorial, we review some of the most common tests for the convergence of an infinite series $$ \\sum_{k=0}^{\\infty} a_k = a_0 + a_1 + a_2 + \\cdots $$ The proofs or these tests are interesting, so we urge you to look them up in&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-174","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/174","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=174"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/174\/revisions"}],"predecessor-version":[{"id":1092,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/174\/revisions\/1092"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=174"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}