{"id":907,"date":"2025-06-20T17:21:28","date_gmt":"2025-06-20T17:21:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=907"},"modified":"2025-07-29T17:56:06","modified_gmt":"2025-07-29T17:56:06","slug":"sequences-and-series-foundations-cheat-sheet-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/sequences-and-series-foundations-cheat-sheet-2\/","title":{"raw":"Power Series and Applications: Cheat Sheet","rendered":"Power Series and Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Introduction to Power Series<\/strong>\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<strong>Operations with Power Series<\/strong>\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<strong>Taylor and Maclaurin Series<\/strong>\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<strong>Applications of Series<\/strong>\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<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 \t<li><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><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>\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<\/dt>\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>\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<\/dt>\r\n \t<dt>\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<\/dt>\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<dl id=\"fs-id1167023915744\">\r\n \t<dt>\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<\/dt>\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":"<h2>Essential Concepts<\/h2>\n<p><strong>Introduction to Power Series<\/strong><\/p>\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<p><strong>Operations with Power Series<\/strong><\/p>\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<p><strong>Taylor and Maclaurin Series<\/strong><\/p>\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<p><strong>Applications of Series<\/strong><\/p>\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<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<li><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<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>\n<\/dt>\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<p> \tinterval of convergence<br \/>\n \tthe set of real numbers <em data-effect=\"italics\">x<\/em> for which a power series converges<\/p>\n<dl id=\"fs-id1170571642090\">\n<dt>\n<\/dt>\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-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<p> \tpower series<br \/>\n \ta 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]<\/p>\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<dl id=\"fs-id1167023915744\">\n<dt>\n<\/dt>\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<p> \tterm-by-term differentiation of a power series<br \/>\n \ta 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]<\/p>\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","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":673,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/907"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/907\/revisions"}],"predecessor-version":[{"id":1658,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/907\/revisions\/1658"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/673"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/907\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=907"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=907"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=907"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}