{"id":710,"date":"2021-05-10T19:08:30","date_gmt":"2021-05-10T19:08:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=710"},"modified":"2021-11-17T23:41:53","modified_gmt":"2021-11-17T23:41:53","slug":"summary-of-working-with-taylor-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-working-with-taylor-series\/","title":{"raw":"Summary of Working with Taylor Series","rendered":"Summary of Working with Taylor Series"},"content":{"raw":"<section id=\"fs-id1167023785990\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167023785997\" data-bullet-style=\"bullet\">\r\n \t<li>The binomial series is the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]. It converges for [latex]|x|&lt;1[\/latex].<\/li>\r\n \t<li>Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.<\/li>\r\n \t<li>Power series can be used to solve differential equations.<\/li>\r\n \t<li>Taylor series can be used to help approximate integrals that cannot be evaluated by other means.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1167023801749\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div style=\"text-align: left;\" data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167023846563\">\r\n \t<dt>binomial series<\/dt>\r\n \t<dd id=\"fs-id1167023846568\">the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]; it is given by<span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\left(1+x\\right)}^{r}=\\displaystyle\\sum _{n=0}^{\\infty }\\left(\\begin{array}{c}r\\hfill \\\\ n\\hfill \\end{array}\\right){x}^{n}=1+rx+\\frac{r\\left(r - 1\\right)}{2\\text{!}}{x}^{2}+\\cdots +\\frac{r\\left(r - 1\\right)\\cdots \\left(r-n+1\\right)}{n\\text{!}}{x}^{n}+\\cdots [\/latex] for [latex]|x|&lt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167023864772\">\r\n \t<dt>nonelementary integral<\/dt>\r\n \t<dd id=\"fs-id1167023864777\">an integral for which the antiderivative of the integrand cannot be expressed as an elementary function<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1167023785990\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167023785997\" data-bullet-style=\"bullet\">\n<li>The binomial series is the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]. It converges for [latex]|x|<1[\/latex].<\/li>\n<li>Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.<\/li>\n<li>Power series can be used to solve differential equations.<\/li>\n<li>Taylor series can be used to help approximate integrals that cannot be evaluated by other means.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1167023801749\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div style=\"text-align: left;\" data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167023846563\">\n<dt>binomial series<\/dt>\n<dd id=\"fs-id1167023846568\">the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]; it is given by<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\left(1+x\\right)}^{r}=\\displaystyle\\sum _{n=0}^{\\infty }\\left(\\begin{array}{c}r\\hfill \\\\ n\\hfill \\end{array}\\right){x}^{n}=1+rx+\\frac{r\\left(r - 1\\right)}{2\\text{!}}{x}^{2}+\\cdots +\\frac{r\\left(r - 1\\right)\\cdots \\left(r-n+1\\right)}{n\\text{!}}{x}^{n}+\\cdots[\/latex] for [latex]|x|<1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167023864772\">\n<dt>nonelementary integral<\/dt>\n<dd id=\"fs-id1167023864777\">an integral for which the antiderivative of the integrand cannot be expressed as an elementary function<\/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-710\">\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":17,"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-710","chapter","type-chapter","status-publish","hentry"],"part":161,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/710","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\/710\/revisions"}],"predecessor-version":[{"id":1069,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/710\/revisions\/1069"}],"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\/710\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=710"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=710"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=710"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}