{"id":921,"date":"2025-06-20T17:22:51","date_gmt":"2025-06-20T17:22:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=921"},"modified":"2025-08-25T15:51:40","modified_gmt":"2025-08-25T15:51:40","slug":"introduction-to-power-series-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-power-series-learn-it-1\/","title":{"raw":"Introduction to Power Series: Learn It 1","rendered":"Introduction to Power Series: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize power series and when they converge<\/li>\r\n \t<li>Find where a power series converges and where it doesn't<\/li>\r\n \t<li>Use power series to write functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">What is a Power Series?<\/h2>\r\n<p class=\"whitespace-normal break-words\">A <strong>power series<\/strong> is a special type of infinite series where each term contains a variable raised to increasing powers. Think of it as an \"infinite polynomial\" \u2014 instead of stopping at [latex]x^3[\/latex] or [latex]x^{10}[\/latex], the powers keep going forever.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The most basic power series looks like this:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \\cdots[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Here, [latex]x[\/latex] is our variable and the [latex]c_n[\/latex] values are constants called coefficients.<\/p>\r\n<p class=\"whitespace-normal break-words\">You've actually seen a power series before. The geometric series [latex]1 + x + x^2 + x^3 + \\cdots[\/latex] is a power series where all coefficients equal [latex]1[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">This series converges when [latex]|x| &lt; 1[\/latex] and diverges when [latex]|x| \\geq 1[\/latex]. This gives us our first hint that power series don't converge everywhere \u2014 the value of [latex]x[\/latex] matters.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>power series centered at zero<\/h3>\r\nA series of the form\r\n\r\n<center>[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \\cdots[\/latex]<\/center>is called a power series centered at [latex]x = 0[\/latex].\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>power series centered at [latex]a[\/latex]<\/h3>\r\nA series of the form\r\n\r\n<center>[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \\cdots[\/latex]<\/center>is called a power series centered at [latex]x = a[\/latex].\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">We always define [latex]x^0 = 1[\/latex] and [latex]{(x-a)}^{0}=1[\/latex] , even when [latex]x = 0[\/latex] or [latex]x = a[\/latex]. This ensures our series starts with the constant term [latex]c_0[\/latex].<\/section>\r\n<p class=\"whitespace-normal break-words\">Here are several power series to help you recognize the pattern.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\"><strong>Centered at<\/strong> [latex]x = 0[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} n! x^n = 1 + x + 2! x^2 + 3! x^3 + \\cdots[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Centered at<\/strong> [latex]x = 2[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} \\frac{(x-2)^n}{(n+1) \\cdot 3^n} = 1 + \\frac{x-2}{2 \\cdot 3} + \\frac{(x-2)^2}{3 \\cdot 3^2} + \\frac{(x-2)^3}{4 \\cdot 3^3} + \\cdots[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize power series and when they converge<\/li>\n<li>Find where a power series converges and where it doesn&#8217;t<\/li>\n<li>Use power series to write functions<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">What is a Power Series?<\/h2>\n<p class=\"whitespace-normal break-words\">A <strong>power series<\/strong> is a special type of infinite series where each term contains a variable raised to increasing powers. Think of it as an &#8220;infinite polynomial&#8221; \u2014 instead of stopping at [latex]x^3[\/latex] or [latex]x^{10}[\/latex], the powers keep going forever.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The most basic power series looks like this:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \\cdots[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Here, [latex]x[\/latex] is our variable and the [latex]c_n[\/latex] values are constants called coefficients.<\/p>\n<p class=\"whitespace-normal break-words\">You&#8217;ve actually seen a power series before. The geometric series [latex]1 + x + x^2 + x^3 + \\cdots[\/latex] is a power series where all coefficients equal [latex]1[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">This series converges when [latex]|x| < 1[\/latex] and diverges when [latex]|x| \\geq 1[\/latex]. This gives us our first hint that power series don&#8217;t converge everywhere \u2014 the value of [latex]x[\/latex] matters.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>power series centered at zero<\/h3>\n<p>A series of the form<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \\cdots[\/latex]<\/div>\n<p>is called a power series centered at [latex]x = 0[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>power series centered at [latex]a[\/latex]<\/h3>\n<p>A series of the form<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \\cdots[\/latex]<\/div>\n<p>is called a power series centered at [latex]x = a[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">We always define [latex]x^0 = 1[\/latex] and [latex]{(x-a)}^{0}=1[\/latex] , even when [latex]x = 0[\/latex] or [latex]x = a[\/latex]. This ensures our series starts with the constant term [latex]c_0[\/latex].<\/section>\n<p class=\"whitespace-normal break-words\">Here are several power series to help you recognize the pattern.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\"><strong>Centered at<\/strong> [latex]x = 0[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} n! x^n = 1 + x + 2! x^2 + 3! x^3 + \\cdots[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Centered at<\/strong> [latex]x = 2[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\sum_{n=0}^{\\infty} \\frac{(x-2)^n}{(n+1) \\cdot 3^n} = 1 + \\frac{x-2}{2 \\cdot 3} + \\frac{(x-2)^2}{3 \\cdot 3^2} + \\frac{(x-2)^3}{4 \\cdot 3^3} + \\cdots[\/latex]<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"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\/921"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/921\/revisions"}],"predecessor-version":[{"id":1958,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/921\/revisions\/1958"}],"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\/921\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=921"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=921"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=921"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}