{"id":698,"date":"2021-05-10T19:05:04","date_gmt":"2021-05-10T19:05:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=698"},"modified":"2021-11-17T03:05:39","modified_gmt":"2021-11-17T03:05:39","slug":"summary-of-ratio-and-root-tests","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-ratio-and-root-tests\/","title":{"raw":"Summary of Ratio and Root Tests","rendered":"Summary of Ratio and Root Tests"},"content":{"raw":"<section id=\"fs-id1169739252731\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169739252738\" data-bullet-style=\"bullet\">\r\n \t<li>For the ratio test, we consider<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169739252749\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nIf [latex]\\rho &lt;1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho &gt;1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. This test is useful for series whose terms involve factorials.<\/li>\r\n \t<li>For the root test, we consider<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169739206929\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nIf [latex]\\rho &lt;1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho &gt;1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. The root test is useful for series whose terms involve powers.<\/li>\r\n \t<li>For a series that is similar to a geometric series or [latex]p-\\text{series,}[\/latex] consider one of the comparison tests.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169736790185\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169736845011\">\r\n \t<dt>ratio test<\/dt>\r\n \t<dd id=\"fs-id1169736845016\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with nonzero terms, let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex]; if [latex]0\\le \\rho &lt;1[\/latex], the series converges absolutely; if [latex]\\rho &gt;1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736845132\">\r\n \t<dt>root test<\/dt>\r\n \t<dd id=\"fs-id1169736845137\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex], let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex]; if [latex]0\\le \\rho &lt;1[\/latex], the series converges absolutely; if [latex]\\rho &gt;1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169739252731\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169739252738\" data-bullet-style=\"bullet\">\n<li>For the ratio test, we consider<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169739252749\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nIf [latex]\\rho <1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho >1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. This test is useful for series whose terms involve factorials.<\/li>\n<li>For the root test, we consider<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169739206929\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nIf [latex]\\rho <1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho >1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. The root test is useful for series whose terms involve powers.<\/li>\n<li>For a series that is similar to a geometric series or [latex]p-\\text{series,}[\/latex] consider one of the comparison tests.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1169736790185\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169736845011\">\n<dt>ratio test<\/dt>\n<dd id=\"fs-id1169736845016\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with nonzero terms, let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex]; if [latex]0\\le \\rho <1[\/latex], the series converges absolutely; if [latex]\\rho >1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736845132\">\n<dt>root test<\/dt>\n<dd id=\"fs-id1169736845137\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex], let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex]; if [latex]0\\le \\rho <1[\/latex], the series converges absolutely; if [latex]\\rho >1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/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-698\">\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":27,"template":"","meta":{"_candela_citation":"{\"1\":{\"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 https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-698","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/698","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/698\/revisions"}],"predecessor-version":[{"id":1061,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/698\/revisions\/1061"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/698\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=698"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=698"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=698"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=698"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}