{"id":708,"date":"2021-05-10T19:08:08","date_gmt":"2021-05-10T19:08:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=708"},"modified":"2021-11-17T23:38:56","modified_gmt":"2021-11-17T23:38:56","slug":"summary-of-properties-of-power-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-properties-of-power-series\/","title":{"raw":"Summary of Properties of Power Series","rendered":"Summary of Properties of Power Series"},"content":{"raw":"<section id=\"fs-id1167023733746\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167023733753\" data-bullet-style=\"bullet\">\r\n \t<li>Given two power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}[\/latex] and [latex]\\displaystyle\\sum _{n=0}^{\\infty }{d}_{n}{x}^{n}[\/latex] that converge to functions <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">g<\/em> on a common interval <em data-effect=\"italics\">I<\/em>, the sum and difference of the two series converge to [latex]f\\pm g[\/latex], respectively, on <em data-effect=\"italics\">I<\/em>. In addition, for any real number <em data-effect=\"italics\">b<\/em> and integer [latex]m\\ge 0[\/latex], the series [latex]\\displaystyle\\sum _{n=0}^{\\infty }b{x}^{m}{c}_{n}{x}^{n}[\/latex] converges to [latex]b{x}^{m}f\\left(x\\right)[\/latex] and the series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(b{x}^{m}\\right)}^{n}[\/latex] converges to [latex]f\\left(b{x}^{m}\\right)[\/latex] whenever <em data-effect=\"italics\">bx<sup>m<\/sup><\/em> is in the interval <em data-effect=\"italics\">I<\/em>.<\/li>\r\n \t<li>Given two power series that converge on an interval [latex]\\left(\\text{-}R,R\\right)[\/latex], the Cauchy product of the two power series converges on the interval [latex]\\left(\\text{-}R,R\\right)[\/latex].<\/li>\r\n \t<li>Given a power series that converges to a function <em data-effect=\"italics\">f<\/em> on an interval [latex]\\left(\\text{-}R,R\\right)[\/latex], the series can be differentiated term-by-term and the resulting series converges to [latex]{f}^{\\prime }[\/latex] on [latex]\\left(\\text{-}R,R\\right)[\/latex]. The series can also be integrated term-by-term and the resulting series converges to [latex]\\displaystyle\\int f\\left(x\\right)dx[\/latex] on [latex]\\left(\\text{-}R,R\\right)[\/latex].<\/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-id1167023915744\">\r\n \t<dt>term-by-term differentiation of a power series<\/dt>\r\n \t<dd id=\"fs-id1167023915750\">a technique for evaluating the derivative of a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] by evaluating the derivative of each term separately to create the new power series [latex]\\displaystyle\\sum _{n=1}^{\\infty }n{c}_{n}{\\left(x-a\\right)}^{n - 1}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167023915846\">\r\n \t<dt>term-by-term integration of a power series<\/dt>\r\n \t<dd id=\"fs-id1167023915851\">a technique for integrating a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] by integrating each term separately to create the new power series [latex]C+\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}\\frac{{\\left(x-a\\right)}^{n+1}}{n+1}[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1167023733746\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167023733753\" data-bullet-style=\"bullet\">\n<li>Given two power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{x}^{n}[\/latex] and [latex]\\displaystyle\\sum _{n=0}^{\\infty }{d}_{n}{x}^{n}[\/latex] that converge to functions <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">g<\/em> on a common interval <em data-effect=\"italics\">I<\/em>, the sum and difference of the two series converge to [latex]f\\pm g[\/latex], respectively, on <em data-effect=\"italics\">I<\/em>. In addition, for any real number <em data-effect=\"italics\">b<\/em> and integer [latex]m\\ge 0[\/latex], the series [latex]\\displaystyle\\sum _{n=0}^{\\infty }b{x}^{m}{c}_{n}{x}^{n}[\/latex] converges to [latex]b{x}^{m}f\\left(x\\right)[\/latex] and the series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(b{x}^{m}\\right)}^{n}[\/latex] converges to [latex]f\\left(b{x}^{m}\\right)[\/latex] whenever <em data-effect=\"italics\">bx<sup>m<\/sup><\/em> is in the interval <em data-effect=\"italics\">I<\/em>.<\/li>\n<li>Given two power series that converge on an interval [latex]\\left(\\text{-}R,R\\right)[\/latex], the Cauchy product of the two power series converges on the interval [latex]\\left(\\text{-}R,R\\right)[\/latex].<\/li>\n<li>Given a power series that converges to a function <em data-effect=\"italics\">f<\/em> on an interval [latex]\\left(\\text{-}R,R\\right)[\/latex], the series can be differentiated term-by-term and the resulting series converges to [latex]{f}^{\\prime }[\/latex] on [latex]\\left(\\text{-}R,R\\right)[\/latex]. The series can also be integrated term-by-term and the resulting series converges to [latex]\\displaystyle\\int f\\left(x\\right)dx[\/latex] on [latex]\\left(\\text{-}R,R\\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167023915744\">\n<dt>term-by-term differentiation of a power series<\/dt>\n<dd id=\"fs-id1167023915750\">a technique for evaluating the derivative of a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] by evaluating the derivative of each term separately to create the new power series [latex]\\displaystyle\\sum _{n=1}^{\\infty }n{c}_{n}{\\left(x-a\\right)}^{n - 1}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167023915846\">\n<dt>term-by-term integration of a power series<\/dt>\n<dd id=\"fs-id1167023915851\">a technique for integrating a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}{\\left(x-a\\right)}^{n}[\/latex] by integrating each term separately to create the new power series [latex]C+\\displaystyle\\sum _{n=0}^{\\infty }{c}_{n}\\frac{{\\left(x-a\\right)}^{n+1}}{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-708\">\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":9,"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-708","chapter","type-chapter","status-publish","hentry"],"part":161,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/708","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\/708\/revisions"}],"predecessor-version":[{"id":1065,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/708\/revisions\/1065"}],"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\/708\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=708"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=708"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=708"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}