{"id":694,"date":"2021-05-10T19:04:18","date_gmt":"2021-05-10T19:04:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=694"},"modified":"2021-11-17T03:01:24","modified_gmt":"2021-11-17T03:01:24","slug":"summary-of-infinite-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-infinite-series\/","title":{"raw":"Summary of Infinite Series","rendered":"Summary of Infinite Series"},"content":{"raw":"<section id=\"fs-id1169737174590\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169737174597\" data-bullet-style=\"bullet\">\r\n \t<li>Given the infinite series<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169737174607\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots [\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nand the corresponding sequence of partial sums [latex]\\left\\{{S}_{k}\\right\\}[\/latex] where<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169737174681\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex],<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nthe series converges if and only if the sequence [latex]\\left\\{{S}_{k}\\right\\}[\/latex] converges.<\/li>\r\n \t<li>The geometric series [latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}[\/latex] converges if [latex]|r|&lt;1[\/latex] and diverges if [latex]|r|\\ge 1[\/latex]. For [latex]|r|&lt;1[\/latex], <span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169737392709\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=\\frac{a}{1-r}[\/latex].<\/div><\/li>\r\n \t<li>The harmonic series<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169737392766\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots [\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\ndiverges.<\/li>\r\n \t<li>A series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\left[{b}_{n}-{b}_{n+1}\\right]=\\left[{b}_{1}-{b}_{2}\\right]+\\left[{b}_{2}-{b}_{3}\\right]+\\left[{b}_{3}-{b}_{4}\\right]+\\cdots +\\left[{b}_{n}-{b}_{n+1}\\right]+\\cdots [\/latex] <span data-type=\"newline\">\r\n<\/span>\r\nis a telescoping series. The [latex]k\\text{th}[\/latex] partial sum of this series is given by [latex]{S}_{k}={b}_{1}-{b}_{k+1}[\/latex]. The series will converge if and only if [latex]\\underset{k\\to \\infty }{\\text{lim}}{b}_{k+1}[\/latex] exists. In that case,<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169737895513\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\left[{b}_{n}-{b}_{n+1}\\right]={b}_{1}-\\underset{k\\to \\infty }{\\text{lim}}\\left({b}_{k+1}\\right)[\/latex].<\/div><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169737169427\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169737169434\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Harmonic series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\cdots [\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Sum of a geometric series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=\\frac{a}{1-r}\\text{ for }|r|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1169737300764\" class=\"section-exercises\" data-depth=\"1\">\r\n<div id=\"fs-id1169737904556\" data-type=\"exercise\"><\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169737160426\">\r\n \t<dt>convergence of a series<\/dt>\r\n \t<dd id=\"fs-id1169737160431\">a series converges if the sequence of partial sums for that series converges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169737160435\">\r\n \t<dt>divergence of a series<\/dt>\r\n \t<dd id=\"fs-id1169737160440\">a series diverges if the sequence of partial sums for that series diverges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169737160445\">\r\n \t<dt>geometric series<\/dt>\r\n \t<dd id=\"fs-id1169737160450\">a geometric series is a series that can be written in the form<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169736694662\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\\cdots [\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738078130\">\r\n \t<dt>harmonic series<\/dt>\r\n \t<dd id=\"fs-id1169738078136\">the harmonic series takes the form<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169736694749\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots [\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738078190\">\r\n \t<dt>infinite series<\/dt>\r\n \t<dd id=\"fs-id1169738078195\">an infinite series is an expression of the form<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169736893183\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{a}_{1}+{a}_{2}+{a}_{3}+\\cdots =\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738078254\">\r\n \t<dt>partial sum<\/dt>\r\n \t<dd id=\"fs-id1169738078259\">the [latex]k\\text{th}[\/latex] partial sum of the infinite series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is the finite sum<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169739133142\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738249026\">\r\n \t<dt>telescoping series<\/dt>\r\n \t<dd id=\"fs-id1169738249032\">a telescoping series is one in which most of the terms cancel in each of the partial sums<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169737174590\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169737174597\" data-bullet-style=\"bullet\">\n<li>Given the infinite series<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169737174607\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nand the corresponding sequence of partial sums [latex]\\left\\{{S}_{k}\\right\\}[\/latex] where<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169737174681\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex],<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nthe series converges if and only if the sequence [latex]\\left\\{{S}_{k}\\right\\}[\/latex] converges.<\/li>\n<li>The geometric series [latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}[\/latex] converges if [latex]|r|<1[\/latex] and diverges if [latex]|r|\\ge 1[\/latex]. For [latex]|r|<1[\/latex], <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169737392709\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=\\frac{a}{1-r}[\/latex].<\/div>\n<\/li>\n<li>The harmonic series<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169737392766\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\ndiverges.<\/li>\n<li>A series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\left[{b}_{n}-{b}_{n+1}\\right]=\\left[{b}_{1}-{b}_{2}\\right]+\\left[{b}_{2}-{b}_{3}\\right]+\\left[{b}_{3}-{b}_{4}\\right]+\\cdots +\\left[{b}_{n}-{b}_{n+1}\\right]+\\cdots[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\nis a telescoping series. The [latex]k\\text{th}[\/latex] partial sum of this series is given by [latex]{S}_{k}={b}_{1}-{b}_{k+1}[\/latex]. The series will converge if and only if [latex]\\underset{k\\to \\infty }{\\text{lim}}{b}_{k+1}[\/latex] exists. In that case,<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169737895513\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\left[{b}_{n}-{b}_{n+1}\\right]={b}_{1}-\\underset{k\\to \\infty }{\\text{lim}}\\left({b}_{k+1}\\right)[\/latex].<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1169737169427\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169737169434\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Harmonic series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\cdots[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Sum of a geometric series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=\\frac{a}{1-r}\\text{ for }|r|<1[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1169737300764\" class=\"section-exercises\" data-depth=\"1\">\n<div id=\"fs-id1169737904556\" data-type=\"exercise\"><\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169737160426\">\n<dt>convergence of a series<\/dt>\n<dd id=\"fs-id1169737160431\">a series converges if the sequence of partial sums for that series converges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169737160435\">\n<dt>divergence of a series<\/dt>\n<dd id=\"fs-id1169737160440\">a series diverges if the sequence of partial sums for that series diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169737160445\">\n<dt>geometric series<\/dt>\n<dd id=\"fs-id1169737160450\">a geometric series is a series that can be written in the form<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169736694662\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\\cdots[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738078130\">\n<dt>harmonic series<\/dt>\n<dd id=\"fs-id1169738078136\">the harmonic series takes the form<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169736694749\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738078190\">\n<dt>infinite series<\/dt>\n<dd id=\"fs-id1169738078195\">an infinite series is an expression of the form<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169736893183\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{a}_{1}+{a}_{2}+{a}_{3}+\\cdots =\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738078254\">\n<dt>partial sum<\/dt>\n<dd id=\"fs-id1169738078259\">the [latex]k\\text{th}[\/latex] partial sum of the infinite series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is the finite sum<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169739133142\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738249026\">\n<dt>telescoping series<\/dt>\n<dd id=\"fs-id1169738249032\">a telescoping series is one in which most of the terms cancel in each of the partial sums<\/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-694\">\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":11,"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-694","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/694","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\/694\/revisions"}],"predecessor-version":[{"id":1238,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/694\/revisions\/1238"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/694\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=694"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=694"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=694"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}