{"id":709,"date":"2021-05-10T19:08:17","date_gmt":"2021-05-10T19:08:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=709"},"modified":"2021-11-17T23:40:14","modified_gmt":"2021-11-17T23:40:14","slug":"summary-of-taylor-and-maclaurin-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-taylor-and-maclaurin-series\/","title":{"raw":"Summary of Taylor and Maclaurin Series","rendered":"Summary of Taylor and Maclaurin Series"},"content":{"raw":"<section id=\"fs-id1167025235824\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167025235831\" data-bullet-style=\"bullet\">\r\n \t<li>Taylor polynomials are used to approximate functions near a value [latex]x=a[\/latex]. Maclaurin polynomials are Taylor polynomials at [latex]x=0[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">n<\/em>th degree Taylor polynomials for a function [latex]f[\/latex] are the partial sums of the Taylor series for [latex]f[\/latex].<\/li>\r\n \t<li>If a function [latex]f[\/latex] has a power series representation at [latex]x=a[\/latex], then it is given by its Taylor series at [latex]x=a[\/latex].<\/li>\r\n \t<li>A Taylor series for [latex]f[\/latex] converges to [latex]f[\/latex] if and only if [latex]\\underset{n\\to \\infty }{\\text{lim}}{R}_{n}\\left(x\\right)=0[\/latex] where [latex]{R}_{n}\\left(x\\right)=f\\left(x\\right)-{p}_{n}\\left(x\\right)[\/latex].<\/li>\r\n \t<li>The Taylor series for <em data-effect=\"italics\">e<sup>x<\/sup><\/em>, [latex]\\sin{x}[\/latex], and [latex]\\cos{x}[\/latex] converge to the respective functions for all real <em data-effect=\"italics\">x<\/em>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1167025236045\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167025236052\" data-bullet-style=\"bullet\">\r\n \t<li style=\"text-align: left;\"><strong data-effect=\"bold\">Taylor series for the function [latex]f[\/latex] at the point<\/strong> [latex]x=a[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}=f\\left(a\\right)+{f}^{\\prime }\\left(a\\right)\\left(x-a\\right)+\\frac{f^{\\prime\\prime}\\left(a\\right)}{2\\text{!}}{\\left(x-a\\right)}^{2}+\\cdots +\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}+\\cdots [\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div style=\"text-align: left;\" data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167025150975\">\r\n \t<dt>Maclaurin polynomial<\/dt>\r\n \t<dd id=\"fs-id1167025150980\">a Taylor polynomial centered at 0; the [latex]n[\/latex]th Taylor polynomial for [latex]f[\/latex] at 0 is the [latex]n[\/latex]th Maclaurin polynomial for [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167025151005\">\r\n \t<dt>Maclaurin series<\/dt>\r\n \t<dd id=\"fs-id1167025151011\">a Taylor series for a function [latex]f[\/latex] at [latex]x=0[\/latex] is known as a Maclaurin series for [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167025151035\">\r\n \t<dt>Taylor polynomials<\/dt>\r\n \t<dd id=\"fs-id1167025151040\">the [latex]n[\/latex]th Taylor polynomial for [latex]f[\/latex] at [latex]x=a[\/latex] is [latex]{p}_{n}\\left(x\\right)=f\\left(a\\right)+{f}^{\\prime }\\left(a\\right)\\left(x-a\\right)+\\frac{f^{\\prime\\prime}\\left(a\\right)}{2\\text{!}}{\\left(x-a\\right)}^{2}+\\cdots +\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167025001267\">\r\n \t<dt>Taylor series<\/dt>\r\n \t<dd id=\"fs-id1167025001272\">a power series at [latex]a[\/latex] that converges to a function [latex]f[\/latex] on some open interval containing [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167025001291\">\r\n \t<dt>Taylor\u2019s theorem with remainder<\/dt>\r\n \t<dd id=\"fs-id1167025001296\">for a function [latex]f[\/latex] and the <em data-effect=\"italics\">n<\/em>th Taylor polynomial for [latex]f[\/latex] at [latex]x=a[\/latex], the remainder [latex]{R}_{n}\\left(x\\right)=f\\left(x\\right)-{p}_{n}\\left(x\\right)[\/latex] satisfies [latex]{R}_{n}\\left(x\\right)=\\frac{{f}^{\\left(n+1\\right)}\\left(c\\right)}{\\left(n+1\\right)\\text{!}}{\\left(x-a\\right)}^{n+1}[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\nfor some [latex]c[\/latex] between [latex]x[\/latex] and [latex]a[\/latex]; if there exists an interval [latex]I[\/latex] containing [latex]a[\/latex] and a real number [latex]M[\/latex] such that [latex]|{f}^{\\left(n+1\\right)}\\left(x\\right)|\\le M[\/latex] for all [latex]x[\/latex] in [latex]I[\/latex], then [latex]|{R}_{n}\\left(x\\right)|\\le \\frac{M}{\\left(n+1\\right)\\text{!}}{|x-a|}^{n+1}[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1167025235824\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167025235831\" data-bullet-style=\"bullet\">\n<li>Taylor polynomials are used to approximate functions near a value [latex]x=a[\/latex]. Maclaurin polynomials are Taylor polynomials at [latex]x=0[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">n<\/em>th degree Taylor polynomials for a function [latex]f[\/latex] are the partial sums of the Taylor series for [latex]f[\/latex].<\/li>\n<li>If a function [latex]f[\/latex] has a power series representation at [latex]x=a[\/latex], then it is given by its Taylor series at [latex]x=a[\/latex].<\/li>\n<li>A Taylor series for [latex]f[\/latex] converges to [latex]f[\/latex] if and only if [latex]\\underset{n\\to \\infty }{\\text{lim}}{R}_{n}\\left(x\\right)=0[\/latex] where [latex]{R}_{n}\\left(x\\right)=f\\left(x\\right)-{p}_{n}\\left(x\\right)[\/latex].<\/li>\n<li>The Taylor series for <em data-effect=\"italics\">e<sup>x<\/sup><\/em>, [latex]\\sin{x}[\/latex], and [latex]\\cos{x}[\/latex] converge to the respective functions for all real <em data-effect=\"italics\">x<\/em>.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1167025236045\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167025236052\" data-bullet-style=\"bullet\">\n<li style=\"text-align: left;\"><strong data-effect=\"bold\">Taylor series for the function [latex]f[\/latex] at the point<\/strong> [latex]x=a[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}=f\\left(a\\right)+{f}^{\\prime }\\left(a\\right)\\left(x-a\\right)+\\frac{f^{\\prime\\prime}\\left(a\\right)}{2\\text{!}}{\\left(x-a\\right)}^{2}+\\cdots +\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}+\\cdots[\/latex]<\/li>\n<\/ul>\n<\/section>\n<div style=\"text-align: left;\" data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167025150975\">\n<dt>Maclaurin polynomial<\/dt>\n<dd id=\"fs-id1167025150980\">a Taylor polynomial centered at 0; the [latex]n[\/latex]th Taylor polynomial for [latex]f[\/latex] at 0 is the [latex]n[\/latex]th Maclaurin polynomial for [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167025151005\">\n<dt>Maclaurin series<\/dt>\n<dd id=\"fs-id1167025151011\">a Taylor series for a function [latex]f[\/latex] at [latex]x=0[\/latex] is known as a Maclaurin series for [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167025151035\">\n<dt>Taylor polynomials<\/dt>\n<dd id=\"fs-id1167025151040\">the [latex]n[\/latex]th Taylor polynomial for [latex]f[\/latex] at [latex]x=a[\/latex] is [latex]{p}_{n}\\left(x\\right)=f\\left(a\\right)+{f}^{\\prime }\\left(a\\right)\\left(x-a\\right)+\\frac{f^{\\prime\\prime}\\left(a\\right)}{2\\text{!}}{\\left(x-a\\right)}^{2}+\\cdots +\\frac{{f}^{\\left(n\\right)}\\left(a\\right)}{n\\text{!}}{\\left(x-a\\right)}^{n}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167025001267\">\n<dt>Taylor series<\/dt>\n<dd id=\"fs-id1167025001272\">a power series at [latex]a[\/latex] that converges to a function [latex]f[\/latex] on some open interval containing [latex]a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167025001291\">\n<dt>Taylor\u2019s theorem with remainder<\/dt>\n<dd id=\"fs-id1167025001296\">for a function [latex]f[\/latex] and the <em data-effect=\"italics\">n<\/em>th Taylor polynomial for [latex]f[\/latex] at [latex]x=a[\/latex], the remainder [latex]{R}_{n}\\left(x\\right)=f\\left(x\\right)-{p}_{n}\\left(x\\right)[\/latex] satisfies [latex]{R}_{n}\\left(x\\right)=\\frac{{f}^{\\left(n+1\\right)}\\left(c\\right)}{\\left(n+1\\right)\\text{!}}{\\left(x-a\\right)}^{n+1}[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\nfor some [latex]c[\/latex] between [latex]x[\/latex] and [latex]a[\/latex]; if there exists an interval [latex]I[\/latex] containing [latex]a[\/latex] and a real number [latex]M[\/latex] such that [latex]|{f}^{\\left(n+1\\right)}\\left(x\\right)|\\le M[\/latex] for all [latex]x[\/latex] in [latex]I[\/latex], then [latex]|{R}_{n}\\left(x\\right)|\\le \\frac{M}{\\left(n+1\\right)\\text{!}}{|x-a|}^{n+1}[\/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-709\">\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":13,"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-709","chapter","type-chapter","status-publish","hentry"],"part":161,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/709","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":5,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/709\/revisions"}],"predecessor-version":[{"id":1874,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/709\/revisions\/1874"}],"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\/709\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=709"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=709"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=709"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}