{"id":706,"date":"2021-05-10T19:08:00","date_gmt":"2021-05-10T19:08:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=706"},"modified":"2021-11-17T23:36:37","modified_gmt":"2021-11-17T23:36:37","slug":"summary-of-power-series-and-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-power-series-and-functions\/","title":{"raw":"Summary of Power Series and Functions","rendered":"Summary of Power Series and Functions"},"content":{"raw":"<section id=\"fs-id1170571546058\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571781586\" data-bullet-style=\"bullet\">\r\n \t<li>For a power series centered at [latex]x=a[\/latex], one of the following three properties hold:\r\n<ol id=\"fs-id1170571781602\" type=\"i\">\r\n \t<li>The power series converges only at [latex]x=a[\/latex]. In this case, we say that the radius of convergence is [latex]R=0[\/latex].<\/li>\r\n \t<li>The power series converges for all real numbers <em data-effect=\"italics\">x<\/em>. In this case, we say that the radius of convergence is [latex]R=\\infty [\/latex].<\/li>\r\n \t<li>There is a real number <em data-effect=\"italics\">R<\/em> such that the series converges for [latex]|x-a|&lt;R[\/latex] and diverges for [latex]|x-a|&gt;R[\/latex]. In this case, the radius of convergence is <em data-effect=\"italics\">R<\/em>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>If a power series converges on a finite interval, the series may or may not converge at the endpoints.<\/li>\r\n \t<li>The ratio test may often be used to determine the radius of convergence.<\/li>\r\n \t<li>The geometric series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{x}^{n}=\\frac{1}{1-x}[\/latex] for [latex]|x|&lt;1[\/latex] allows us to represent certain functions using geometric series.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170572420968\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572420975\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Power series centered at<\/strong> [latex]x=0[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\\cdots [\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Power series centered at<\/strong> [latex]x=a[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}={c}_{0}+{c}_{1}\\left(x-a\\right)+{c}_{2}{\\left(x-a\\right)}^{2}+\\cdots [\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1170572516483\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170571642075\">\r\n \t<dt>interval of convergence<\/dt>\r\n \t<dd id=\"fs-id1170571642080\">the set of real numbers <em data-effect=\"italics\">x<\/em> for which a power series converges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571642090\">\r\n \t<dt>power series<\/dt>\r\n \t<dd id=\"fs-id1170571642095\">a series of the form [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}[\/latex] is a power series centered at [latex]x=0[\/latex]; a series of the form [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] is a power series centered at [latex]x=a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571674101\">\r\n \t<dt>radius of convergence<\/dt>\r\n \t<dd id=\"fs-id1170571674107\">if there exists a real number [latex]R&gt;0[\/latex] such that a power series centered at [latex]x=a[\/latex] converges for [latex]|x-a|&lt;R[\/latex] and diverges for [latex]|x-a|&gt;R[\/latex], then <em data-effect=\"italics\">R<\/em> is the radius of convergence; if the power series only converges at [latex]x=a[\/latex], the radius of convergence is [latex]R=0[\/latex]; if the power series converges for all real numbers <em data-effect=\"italics\">x<\/em>, the radius of convergence is [latex]R=\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170571546058\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571781586\" data-bullet-style=\"bullet\">\n<li>For a power series centered at [latex]x=a[\/latex], one of the following three properties hold:\n<ol id=\"fs-id1170571781602\" type=\"i\">\n<li>The power series converges only at [latex]x=a[\/latex]. In this case, we say that the radius of convergence is [latex]R=0[\/latex].<\/li>\n<li>The power series converges for all real numbers <em data-effect=\"italics\">x<\/em>. In this case, we say that the radius of convergence is [latex]R=\\infty[\/latex].<\/li>\n<li>There is a real number <em data-effect=\"italics\">R<\/em> such that the series converges for [latex]|x-a|<R[\/latex] and diverges for [latex]|x-a|>R[\/latex]. In this case, the radius of convergence is <em data-effect=\"italics\">R<\/em>.<\/li>\n<\/ol>\n<\/li>\n<li>If a power series converges on a finite interval, the series may or may not converge at the endpoints.<\/li>\n<li>The ratio test may often be used to determine the radius of convergence.<\/li>\n<li>The geometric series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{x}^{n}=\\frac{1}{1-x}[\/latex] for [latex]|x|<1[\/latex] allows us to represent certain functions using geometric series.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170572420968\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572420975\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Power series centered at<\/strong> [latex]x=0[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\\cdots[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Power series centered at<\/strong> [latex]x=a[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}={c}_{0}+{c}_{1}\\left(x-a\\right)+{c}_{2}{\\left(x-a\\right)}^{2}+\\cdots[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1170572516483\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170571642075\">\n<dt>interval of convergence<\/dt>\n<dd id=\"fs-id1170571642080\">the set of real numbers <em data-effect=\"italics\">x<\/em> for which a power series converges<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571642090\">\n<dt>power series<\/dt>\n<dd id=\"fs-id1170571642095\">a series of the form [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}[\/latex] is a power series centered at [latex]x=0[\/latex]; a series of the form [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] is a power series centered at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571674101\">\n<dt>radius of convergence<\/dt>\n<dd id=\"fs-id1170571674107\">if there exists a real number [latex]R>0[\/latex] such that a power series centered at [latex]x=a[\/latex] converges for [latex]|x-a|<R[\/latex] and diverges for [latex]|x-a|>R[\/latex], then <em data-effect=\"italics\">R<\/em> is the radius of convergence; if the power series only converges at [latex]x=a[\/latex], the radius of convergence is [latex]R=0[\/latex]; if the power series converges for all real numbers <em data-effect=\"italics\">x<\/em>, the radius of convergence is [latex]R=\\infty[\/latex]<\/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-706\">\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":5,"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-706","chapter","type-chapter","status-publish","hentry"],"part":161,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706","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\/706\/revisions"}],"predecessor-version":[{"id":1063,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions\/1063"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/161"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=706"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=706"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=706"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}