{"id":696,"date":"2021-05-10T19:04:41","date_gmt":"2021-05-10T19:04:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=696"},"modified":"2021-11-17T03:03:50","modified_gmt":"2021-11-17T03:03:50","slug":"summary-of-comparison-tests","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-comparison-tests\/","title":{"raw":"Summary of Comparison Tests","rendered":"Summary of Comparison Tests"},"content":{"raw":"<section id=\"fs-id1169736777480\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169739110996\" data-bullet-style=\"bullet\">\r\n \t<li>The comparison tests are used to determine convergence or divergence of series with positive terms.<\/li>\r\n \t<li>When using the comparison tests, a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is often compared to a geometric or <em data-effect=\"italics\">p<\/em>-series.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169736778029\">\r\n \t<dt>comparison test<\/dt>\r\n \t<dd id=\"fs-id1169736778033\">if [latex]0\\le {a}_{n}\\le {b}_{n}[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges; if [latex]{a}_{n}\\ge {b}_{n}\\ge 0[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736592441\">\r\n \t<dt>limit comparison test<\/dt>\r\n \t<dd id=\"fs-id1169736592446\">suppose [latex]{a}_{n},{b}_{n}\\ge 0[\/latex] for all [latex]n\\ge 1[\/latex]. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to L\\ne 0[\/latex], then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] both converge or both diverge; if [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to 0[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to \\infty [\/latex], and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169736777480\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169739110996\" data-bullet-style=\"bullet\">\n<li>The comparison tests are used to determine convergence or divergence of series with positive terms.<\/li>\n<li>When using the comparison tests, a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is often compared to a geometric or <em data-effect=\"italics\">p<\/em>-series.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169736778029\">\n<dt>comparison test<\/dt>\n<dd id=\"fs-id1169736778033\">if [latex]0\\le {a}_{n}\\le {b}_{n}[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges; if [latex]{a}_{n}\\ge {b}_{n}\\ge 0[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736592441\">\n<dt>limit comparison test<\/dt>\n<dd id=\"fs-id1169736592446\">suppose [latex]{a}_{n},{b}_{n}\\ge 0[\/latex] for all [latex]n\\ge 1[\/latex]. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to L\\ne 0[\/latex], then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] both converge or both diverge; if [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to 0[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to \\infty[\/latex], and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/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-696\">\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":19,"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-696","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/696","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":3,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/696\/revisions"}],"predecessor-version":[{"id":1057,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/696\/revisions\/1057"}],"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\/696\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=696"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=696"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=696"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}