{"id":695,"date":"2021-05-10T19:04:30","date_gmt":"2021-05-10T19:04:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=695"},"modified":"2021-11-17T03:02:53","modified_gmt":"2021-11-17T03:02:53","slug":"summary-of-the-divergence-and-integral-tests","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-divergence-and-integral-tests\/","title":{"raw":"Summary of the Divergence and Integral Tests","rendered":"Summary of the Divergence and Integral Tests"},"content":{"raw":"<section id=\"fs-id1169738052811\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169738153460\" data-bullet-style=\"bullet\">\r\n \t<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges.<\/li>\r\n \t<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=0[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] may converge or diverge.<\/li>\r\n \t<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is a series with positive terms [latex]{a}_{n}[\/latex] and [latex]f[\/latex] is a continuous, decreasing function such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738155262\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{and}{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\neither both converge or both diverge. Furthermore, if [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, then the [latex]N\\text{th}[\/latex] partial sum approximation [latex]{S}_{N}[\/latex] is accurate up to an error [latex]{R}_{N}[\/latex] where [latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">p<\/em>-series [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex] converges if [latex]p&gt;1[\/latex] and diverges if [latex]p\\le 1[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169737162013\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169737162020\" data-bullet-style=\"bullet\">\r\n \t<li style=\"text-align: left;\"><strong data-effect=\"bold\">Divergence test<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\text{If }{a}_{n}\\nrightarrow 0\\text{ as }n\\to \\infty ,\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ diverges}[\/latex].<\/li>\r\n \t<li><strong data-effect=\"bold\"><em data-effect=\"italics\">p<\/em>-series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\sum _{n=1}^{\\infty}} \\dfrac{1}{n^{p}} \\bigg\\{ \\begin{array}{l}\\text{ converges if }p&gt;1\\\\ \\text{ diverges if }p\\le 1\\end{array}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Remainder estimate from the integral test<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1169737430005\" class=\"section-exercises\" data-depth=\"1\">\r\n<div id=\"fs-id1169737214923\" data-type=\"exercise\"><\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169738154867\">\r\n \t<dt>divergence test<\/dt>\r\n \t<dd id=\"fs-id1169738154872\">if [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738154932\">\r\n \t<dt>integral test<\/dt>\r\n \t<dd id=\"fs-id1169738154937\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex], if there exists a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738161782\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\r\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\">either both converge or both diverge<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169737433691\">\r\n \t<dt><em data-effect=\"italics\">p<\/em>-series<\/dt>\r\n \t<dd id=\"fs-id1169737433701\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169737174578\">\r\n \t<dt>remainder estimate<\/dt>\r\n \t<dd id=\"fs-id1169737174583\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex] and a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], the remainder [latex]{R}_{N}=\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}-\\displaystyle\\sum _{n=1}^{N}{a}_{n}[\/latex] satisfies the following estimate:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738066622\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169738052811\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169738153460\" data-bullet-style=\"bullet\">\n<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges.<\/li>\n<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=0[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] may converge or diverge.<\/li>\n<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is a series with positive terms [latex]{a}_{n}[\/latex] and [latex]f[\/latex] is a continuous, decreasing function such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738155262\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{and}{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\neither both converge or both diverge. Furthermore, if [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, then the [latex]N\\text{th}[\/latex] partial sum approximation [latex]{S}_{N}[\/latex] is accurate up to an error [latex]{R}_{N}[\/latex] where [latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">p<\/em>-series [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex] converges if [latex]p>1[\/latex] and diverges if [latex]p\\le 1[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1169737162013\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169737162020\" data-bullet-style=\"bullet\">\n<li style=\"text-align: left;\"><strong data-effect=\"bold\">Divergence test<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\text{If }{a}_{n}\\nrightarrow 0\\text{ as }n\\to \\infty ,\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ diverges}[\/latex].<\/li>\n<li><strong data-effect=\"bold\"><em data-effect=\"italics\">p<\/em>-series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\sum _{n=1}^{\\infty}} \\dfrac{1}{n^{p}} \\bigg\\{ \\begin{array}{l}\\text{ converges if }p>1\\\\ \\text{ diverges if }p\\le 1\\end{array}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Remainder estimate from the integral test<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1169737430005\" class=\"section-exercises\" data-depth=\"1\">\n<div id=\"fs-id1169737214923\" data-type=\"exercise\"><\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169738154867\">\n<dt>divergence test<\/dt>\n<dd id=\"fs-id1169738154872\">if [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738154932\">\n<dt>integral test<\/dt>\n<dd id=\"fs-id1169738154937\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex], if there exists a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738161782\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\">either both converge or both diverge<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169737433691\">\n<dt><em data-effect=\"italics\">p<\/em>-series<\/dt>\n<dd id=\"fs-id1169737433701\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169737174578\">\n<dt>remainder estimate<\/dt>\n<dd id=\"fs-id1169737174583\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex] and a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], the remainder [latex]{R}_{N}=\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}-\\displaystyle\\sum _{n=1}^{N}{a}_{n}[\/latex] satisfies the following estimate:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738066622\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-695\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":416434,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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