{"id":697,"date":"2021-05-10T19:04:51","date_gmt":"2021-05-10T19:04:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=697"},"modified":"2021-11-17T03:04:43","modified_gmt":"2021-11-17T03:04:43","slug":"summary-of-alternating-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-alternating-series\/","title":{"raw":"Summary of Alternating Series","rendered":"Summary of Alternating Series"},"content":{"raw":"<section id=\"fs-id1169737985058\" data-depth=\"1\"><section id=\"fs-id1169738115704\" data-depth=\"2\"><section id=\"fs-id1169737162243\" class=\"key-concepts\" data-depth=\"3\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169737162250\" data-bullet-style=\"bullet\">\r\n \t<li>For an alternating series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex], if [latex]{b}_{k+1}\\le {b}_{k}[\/latex] for all [latex]k[\/latex] and [latex]{b}_{k}\\to 0[\/latex] as [latex]k\\to \\infty [\/latex], the alternating series converges.<\/li>\r\n \t<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169738185039\" class=\"key-equations\" data-depth=\"3\">\r\n<h2 data-type=\"title\">Key Equations<\/h2>\r\n<ul id=\"fs-id1169738185046\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Alternating series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\\cdots \\text{or}[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}=\\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\\cdots [\/latex]<\/li>\r\n<\/ul>\r\n<\/section><\/section><\/section><section id=\"fs-id1169737927676\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\r\n<dl id=\"fs-id1169738193975\">\r\n \t<dt>absolute convergence<\/dt>\r\n \t<dd id=\"fs-id1169738193979\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge absolutely<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738110013\">\r\n \t<dt>alternating series<\/dt>\r\n \t<dd id=\"fs-id1169738110018\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex] or [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}[\/latex], where [latex]{b}_{n}\\ge 0[\/latex], is called an alternating series<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738110115\">\r\n \t<dt>alternating series test<\/dt>\r\n \t<dd id=\"fs-id1169738110120\">for an alternating series of either form, if [latex]{b}_{n+1}\\le {b}_{n}[\/latex] for all integers [latex]n\\ge 1[\/latex] and [latex]{b}_{n}\\to 0[\/latex], then an alternating series converges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738040638\">\r\n \t<dt>conditional convergence<\/dt>\r\n \t<dd id=\"fs-id1169738040642\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, but the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] diverges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge conditionally<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169737985058\" data-depth=\"1\">\n<section id=\"fs-id1169738115704\" data-depth=\"2\">\n<section id=\"fs-id1169737162243\" class=\"key-concepts\" data-depth=\"3\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169737162250\" data-bullet-style=\"bullet\">\n<li>For an alternating series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex], if [latex]{b}_{k+1}\\le {b}_{k}[\/latex] for all [latex]k[\/latex] and [latex]{b}_{k}\\to 0[\/latex] as [latex]k\\to \\infty[\/latex], the alternating series converges.<\/li>\n<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1169738185039\" class=\"key-equations\" data-depth=\"3\">\n<h2 data-type=\"title\">Key Equations<\/h2>\n<ul id=\"fs-id1169738185046\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Alternating series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\\cdots \\text{or}[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}=\\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\\cdots[\/latex]<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<\/section>\n<section id=\"fs-id1169737927676\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1169738193975\">\n<dt>absolute convergence<\/dt>\n<dd id=\"fs-id1169738193979\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge absolutely<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738110013\">\n<dt>alternating series<\/dt>\n<dd id=\"fs-id1169738110018\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex] or [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}[\/latex], where [latex]{b}_{n}\\ge 0[\/latex], is called an alternating series<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738110115\">\n<dt>alternating series test<\/dt>\n<dd id=\"fs-id1169738110120\">for an alternating series of either form, if [latex]{b}_{n+1}\\le {b}_{n}[\/latex] for all integers [latex]n\\ge 1[\/latex] and [latex]{b}_{n}\\to 0[\/latex], then an alternating series converges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738040638\">\n<dt>conditional convergence<\/dt>\n<dd id=\"fs-id1169738040642\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, but the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] diverges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge conditionally<\/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-697\">\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 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