{"id":849,"date":"2025-06-20T17:17:34","date_gmt":"2025-06-20T17:17:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=849"},"modified":"2025-07-29T17:15:16","modified_gmt":"2025-07-29T17:15:16","slug":"sequences-and-series-foundations-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/sequences-and-series-foundations-cheat-sheet\/","title":{"raw":"Sequences and Series Foundations: Cheat Sheet","rendered":"Sequences and Series Foundations: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Sequences and Their Properties<\/strong>\r\n<ul id=\"fs-id1169736857071\" data-bullet-style=\"bullet\">\r\n \t<li>To determine the convergence of a sequence given by an explicit formula [latex]{a}_{n}=f\\left(n\\right)[\/latex], we use the properties of limits for functions.<\/li>\r\n \t<li>If [latex]\\left\\{{a}_{n}\\right\\}[\/latex] and [latex]\\left\\{{b}_{n}\\right\\}[\/latex] are convergent sequences that converge to [latex]A[\/latex] and [latex]B[\/latex], respectively, and [latex]c[\/latex] is any real number, then the sequence [latex]\\left\\{c{a}_{n}\\right\\}[\/latex] converges to [latex]c\\cdot A[\/latex], the sequences [latex]\\left\\{{a}_{n}\\pm {b}_{n}\\right\\}[\/latex] converge to [latex]A\\pm B[\/latex], the sequence [latex]\\left\\{{a}_{n}\\cdot {b}_{n}\\right\\}[\/latex] converges to [latex]A\\cdot B[\/latex], and the sequence [latex]\\left\\{\\frac{{a}_{n}}{{b}_{n}}\\right\\}[\/latex] converges to [latex]\\frac{A}{B}[\/latex], provided [latex]B\\ne 0[\/latex].<\/li>\r\n \t<li>If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.<\/li>\r\n \t<li>If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.<\/li>\r\n \t<li>The geometric sequence [latex]\\left\\{{r}^{n}\\right\\}[\/latex] converges if and only if [latex]|r|&lt;1[\/latex] or [latex]r=1[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Introduction to Series<\/strong>\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]\r\nand the corresponding sequence of partial sums [latex]\\left\\{{S}_{k}\\right\\}[\/latex] where<span data-type=\"newline\">\r\n<\/span><\/div>\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],\r\nthe series converges if and only if the sequence [latex]\\left\\{{S}_{k}\\right\\}[\/latex] converges.<\/div><\/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]\r\ndiverges.<\/div><\/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]\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<strong>The Divergence and Integral Tests<\/strong>\r\n<ul id=\"fs-id1169738153460\" data-bullet-style=\"bullet\">\r\n \t<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges.<\/li>\r\n \t<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=0[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] may converge or diverge.<\/li>\r\n \t<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is a series with positive terms [latex]{a}_{n}[\/latex] and [latex]f[\/latex] is a continuous, decreasing function such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738155262\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{and}{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]\r\neither both converge or both diverge. Furthermore, if [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, then the [latex]N\\text{th}[\/latex] partial sum approximation [latex]{S}_{N}[\/latex] is accurate up to an error [latex]{R}_{N}[\/latex] where [latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex].<\/div><\/li>\r\n \t<li>The <em data-effect=\"italics\">p<\/em>-series [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex] converges if [latex]p&gt;1[\/latex] and diverges if [latex]p\\le 1[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Comparison Tests<\/strong>\r\n<ul id=\"fs-id1169739110996\" data-bullet-style=\"bullet\">\r\n \t<li>The comparison tests are used to determine convergence or divergence of series with positive terms.<\/li>\r\n \t<li>When using the comparison tests, a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is often compared to a geometric or <em data-effect=\"italics\">p<\/em>-series.<\/li>\r\n<\/ul>\r\n<strong>Alternating Series<\/strong>\r\n<ul id=\"fs-id1169737162250\" data-bullet-style=\"bullet\">\r\n \t<li>For an alternating series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex], if [latex]{b}_{k+1}\\le {b}_{k}[\/latex] for all [latex]k[\/latex] and [latex]{b}_{k}\\to 0[\/latex] as [latex]k\\to \\infty [\/latex], the alternating series converges.<\/li>\r\n \t<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges.<\/li>\r\n<\/ul>\r\n<strong>Ratio and Root Tests<\/strong>\r\n<ul id=\"fs-id1169739252738\" data-bullet-style=\"bullet\">\r\n \t<li>For the ratio test, we consider<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169739252749\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nIf [latex]\\rho &lt;1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho &gt;1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. This test is useful for series whose terms involve factorials.<\/li>\r\n \t<li>For the root test, we consider<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169739206929\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nIf [latex]\\rho &lt;1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho &gt;1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. The root test is useful for series whose terms involve powers.<\/li>\r\n \t<li>For a series that is similar to a geometric series or [latex]p-\\text{series,}[\/latex] consider one of the comparison tests.<\/li>\r\n<\/ul>\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 \t<li style=\"text-align: left;\"><strong data-effect=\"bold\">Divergence test<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\text{If }{a}_{n}\\nrightarrow 0\\text{ as }n\\to \\infty ,\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ diverges}[\/latex].<\/li>\r\n \t<li><strong data-effect=\"bold\"><em data-effect=\"italics\">p<\/em>-series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\sum _{n=1}^{\\infty}} \\dfrac{1}{n^{p}} \\bigg\\{ \\begin{array}{l}\\text{ converges if }p&gt;1\\\\ \\text{ diverges if }p\\le 1\\end{array}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Remainder estimate from the integral test<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Alternating series<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\\cdots \\text{or}[\/latex] <span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}=\\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\\cdots [\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169736845011\">\r\n \t<dt>\r\n<dl id=\"fs-id1169739249448\">\r\n \t<dt>\r\n<dl id=\"fs-id1169738193975\">\r\n \t<dt>absolute convergence<\/dt>\r\n \t<dd id=\"fs-id1169738193979\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge absolutely<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738110013\">\r\n \t<dt>alternating series<\/dt>\r\n \t<dd id=\"fs-id1169738110018\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex] or [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}[\/latex], where [latex]{b}_{n}\\ge 0[\/latex], is called an alternating series<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738110115\">\r\n \t<dt>alternating series test<\/dt>\r\n \t<dd id=\"fs-id1169738110120\">for an alternating series of either form, if [latex]{b}_{n+1}\\le {b}_{n}[\/latex] for all integers [latex]n\\ge 1[\/latex] and [latex]{b}_{n}\\to 0[\/latex], then an alternating series converges<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>arithmetic sequence<\/dt>\r\n \t<dd id=\"fs-id1169739249453\">a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739249458\">\r\n \t<dt>bounded above<\/dt>\r\n \t<dd id=\"fs-id1169739249464\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded above if there exists a constant [latex]M[\/latex] such that [latex]{a}_{n}\\le M[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739249504\">\r\n \t<dt>bounded below<\/dt>\r\n \t<dd id=\"fs-id1169739249509\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded below if there exists a constant [latex]M[\/latex] such that [latex]M\\le {a}_{n}[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739249549\">\r\n \t<dt>bounded sequence<\/dt>\r\n \t<dd id=\"fs-id1169739249554\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded if there exists a constant [latex]M[\/latex] such that [latex]|{a}_{n}|\\le M[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739249599\">\r\n \t<dt>\r\n<dl id=\"fs-id1169737160426\">\r\n \t<dt>\r\n<dl id=\"fs-id1169736778029\">\r\n \t<dt>comparison test<\/dt>\r\n \t<dd id=\"fs-id1169736778033\">if [latex]0\\le {a}_{n}\\le {b}_{n}[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges; if [latex]{a}_{n}\\ge {b}_{n}\\ge 0[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1169738040638\">\r\n \t<dt>conditional convergence<\/dt>\r\n \t<dd id=\"fs-id1169738040642\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, but the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] diverges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge conditionally<\/dd>\r\n<\/dl>\r\n<\/dt>\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<\/dt>\r\n \t<dt>convergent sequence<\/dt>\r\n \t<dd id=\"fs-id1169739249604\">a convergent sequence is a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] for which there exists a real number [latex]L[\/latex] such that [latex]{a}_{n}[\/latex] is arbitrarily close to [latex]L[\/latex] as long as [latex]n[\/latex] is sufficiently large<\/dd>\r\n<\/dl>\r\n<dl>\r\n \t<dt>\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<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1169738154867\">\r\n \t<dt>divergence test<\/dt>\r\n \t<dd id=\"fs-id1169738154872\">if [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739249646\">\r\n \t<dt>divergent sequence<\/dt>\r\n \t<dd id=\"fs-id1169736702608\">a sequence that is not convergent is divergent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702613\">\r\n \t<dt>explicit formula<\/dt>\r\n \t<dd id=\"fs-id1169736702618\">a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\\left(n\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702642\">\r\n \t<dt>geometric sequence<\/dt>\r\n \t<dd id=\"fs-id1169736702647\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] in which the ratio [latex]\\frac{{a}_{n+1}}{{a}_{n}}[\/latex] is the same for all positive integers [latex]n[\/latex] is called a geometric sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702692\">\r\n \t<dt>\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<\/dt>\r\n \t<dt>\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<\/dt>\r\n \t<dt>index variable<\/dt>\r\n \t<dd id=\"fs-id1169736702698\">the subscript used to define the terms in a sequence is called the index<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702702\">\r\n \t<dt>\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<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1169738154932\">\r\n \t<dt>integral test<\/dt>\r\n \t<dd id=\"fs-id1169738154937\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex], if there exists a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738161782\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1169736592441\">\r\n \t<dt>limit comparison test<\/dt>\r\n \t<dd id=\"fs-id1169736592446\">suppose [latex]{a}_{n},{b}_{n}\\ge 0[\/latex] for all [latex]n\\ge 1[\/latex]. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to L\\ne 0[\/latex], then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] both converge or both diverge; if [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to 0[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to \\infty [\/latex], and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>limit of a sequence<\/dt>\r\n \t<dd id=\"fs-id1169736702707\">the real number [latex]L[\/latex] to which a sequence converges is called the limit of the sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702717\">\r\n \t<dt>monotone sequence<\/dt>\r\n \t<dd id=\"fs-id1169736702722\">an increasing or decreasing sequence<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\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<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1169737433691\">\r\n \t<dt><em data-effect=\"italics\">p<\/em>-series<\/dt>\r\n \t<dd id=\"fs-id1169737433701\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>ratio test<\/dt>\r\n \t<dd id=\"fs-id1169736845016\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with nonzero terms, let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex]; if [latex]0\\le \\rho &lt;1[\/latex], the series converges absolutely; if [latex]\\rho &gt;1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736845132\">\r\n \t<dt>\r\n<dl id=\"fs-id1169736702727\">\r\n \t<dt>recurrence relation<\/dt>\r\n \t<dd id=\"fs-id1169736702732\">a recurrence relation is a relationship in which a term [latex]{a}_{n}[\/latex] in a sequence is defined in terms of earlier terms in the sequence<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1169737174578\">\r\n \t<dt>remainder estimate<\/dt>\r\n \t<dd id=\"fs-id1169737174583\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex] and a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], the remainder [latex]{R}_{N}=\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}-\\displaystyle\\sum _{n=1}^{N}{a}_{n}[\/latex] satisfies the following estimate:<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169738066622\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx&lt;{R}_{N}&lt;{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>root test<\/dt>\r\n \t<dd id=\"fs-id1169736845137\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex], let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex]; if [latex]0\\le \\rho &lt;1[\/latex], the series converges absolutely; if [latex]\\rho &gt;1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702745\">\r\n \t<dt>sequence<\/dt>\r\n \t<dd id=\"fs-id1169736702750\">an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\\text{,}\\ldots[\/latex] is a sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736702783\">\r\n \t<dt>\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<\/dt>\r\n \t<dt>term<\/dt>\r\n \t<dd id=\"fs-id1169736702788\">the number [latex]{a}_{n}[\/latex] in the sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is called the [latex]n\\text{th}[\/latex] term of the sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736634909\">\r\n \t<dt>unbounded sequence<\/dt>\r\n \t<dd id=\"fs-id1169736634914\">a sequence that is not bounded is called unbounded<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Sequences and Their Properties<\/strong><\/p>\n<ul id=\"fs-id1169736857071\" data-bullet-style=\"bullet\">\n<li>To determine the convergence of a sequence given by an explicit formula [latex]{a}_{n}=f\\left(n\\right)[\/latex], we use the properties of limits for functions.<\/li>\n<li>If [latex]\\left\\{{a}_{n}\\right\\}[\/latex] and [latex]\\left\\{{b}_{n}\\right\\}[\/latex] are convergent sequences that converge to [latex]A[\/latex] and [latex]B[\/latex], respectively, and [latex]c[\/latex] is any real number, then the sequence [latex]\\left\\{c{a}_{n}\\right\\}[\/latex] converges to [latex]c\\cdot A[\/latex], the sequences [latex]\\left\\{{a}_{n}\\pm {b}_{n}\\right\\}[\/latex] converge to [latex]A\\pm B[\/latex], the sequence [latex]\\left\\{{a}_{n}\\cdot {b}_{n}\\right\\}[\/latex] converges to [latex]A\\cdot B[\/latex], and the sequence [latex]\\left\\{\\frac{{a}_{n}}{{b}_{n}}\\right\\}[\/latex] converges to [latex]\\frac{A}{B}[\/latex], provided [latex]B\\ne 0[\/latex].<\/li>\n<li>If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.<\/li>\n<li>If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.<\/li>\n<li>The geometric sequence [latex]\\left\\{{r}^{n}\\right\\}[\/latex] converges if and only if [latex]|r|<1[\/latex] or [latex]r=1[\/latex].<\/li>\n<\/ul>\n<p><strong>Introduction to Series<\/strong><\/p>\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]<br \/>\nand the corresponding sequence of partial sums [latex]\\left\\{{S}_{k}\\right\\}[\/latex] where<span data-type=\"newline\"><br \/>\n<\/span><\/div>\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],<br \/>\nthe series converges if and only if the sequence [latex]\\left\\{{S}_{k}\\right\\}[\/latex] converges.<\/div>\n<\/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]<br \/>\ndiverges.<\/div>\n<\/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]<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<p><strong>The Divergence and Integral Tests<\/strong><\/p>\n<ul id=\"fs-id1169738153460\" data-bullet-style=\"bullet\">\n<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges.<\/li>\n<li>If [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=0[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] may converge or diverge.<\/li>\n<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is a series with positive terms [latex]{a}_{n}[\/latex] and [latex]f[\/latex] is a continuous, decreasing function such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738155262\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{and}{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<br \/>\neither both converge or both diverge. Furthermore, if [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, then the [latex]N\\text{th}[\/latex] partial sum approximation [latex]{S}_{N}[\/latex] is accurate up to an error [latex]{R}_{N}[\/latex] where [latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex].<\/div>\n<\/li>\n<li>The <em data-effect=\"italics\">p<\/em>-series [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex] converges if [latex]p>1[\/latex] and diverges if [latex]p\\le 1[\/latex].<\/li>\n<\/ul>\n<p><strong>Comparison Tests<\/strong><\/p>\n<ul id=\"fs-id1169739110996\" data-bullet-style=\"bullet\">\n<li>The comparison tests are used to determine convergence or divergence of series with positive terms.<\/li>\n<li>When using the comparison tests, a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is often compared to a geometric or <em data-effect=\"italics\">p<\/em>-series.<\/li>\n<\/ul>\n<p><strong>Alternating Series<\/strong><\/p>\n<ul id=\"fs-id1169737162250\" data-bullet-style=\"bullet\">\n<li>For an alternating series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex], if [latex]{b}_{k+1}\\le {b}_{k}[\/latex] for all [latex]k[\/latex] and [latex]{b}_{k}\\to 0[\/latex] as [latex]k\\to \\infty[\/latex], the alternating series converges.<\/li>\n<li>If [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges.<\/li>\n<\/ul>\n<p><strong>Ratio and Root Tests<\/strong><\/p>\n<ul id=\"fs-id1169739252738\" data-bullet-style=\"bullet\">\n<li>For the ratio test, we consider<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169739252749\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nIf [latex]\\rho <1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho >1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. This test is useful for series whose terms involve factorials.<\/li>\n<li>For the root test, we consider<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169739206929\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nIf [latex]\\rho <1[\/latex], the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges absolutely. If [latex]\\rho >1[\/latex], the series diverges. If [latex]\\rho =1[\/latex], the test does not provide any information. The root test is useful for series whose terms involve powers.<\/li>\n<li>For a series that is similar to a geometric series or [latex]p-\\text{series,}[\/latex] consider one of the comparison tests.<\/li>\n<\/ul>\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<li style=\"text-align: left;\"><strong data-effect=\"bold\">Divergence test<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\text{If }{a}_{n}\\nrightarrow 0\\text{ as }n\\to \\infty ,\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ diverges}[\/latex].<\/li>\n<li><strong data-effect=\"bold\"><em data-effect=\"italics\">p<\/em>-series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\sum _{n=1}^{\\infty}} \\dfrac{1}{n^{p}} \\bigg\\{ \\begin{array}{l}\\text{ converges if }p>1\\\\ \\text{ diverges if }p\\le 1\\end{array}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Remainder estimate from the integral test<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Alternating series<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\\cdots \\text{or}[\/latex] <span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}=\\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\\cdots[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169736845011\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt>absolute convergence<\/dt>\n<dd id=\"fs-id1169738193979\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] converges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge absolutely<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738110013\">\n<dt>alternating series<\/dt>\n<dd id=\"fs-id1169738110018\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n+1}{b}_{n}[\/latex] or [latex]\\displaystyle\\sum _{n=1}^{\\infty }{\\left(-1\\right)}^{n}{b}_{n}[\/latex], where [latex]{b}_{n}\\ge 0[\/latex], is called an alternating series<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738110115\">\n<dt>alternating series test<\/dt>\n<dd id=\"fs-id1169738110120\">for an alternating series of either form, if [latex]{b}_{n+1}\\le {b}_{n}[\/latex] for all integers [latex]n\\ge 1[\/latex] and [latex]{b}_{n}\\to 0[\/latex], then an alternating series converges<\/dd>\n<\/dl>\n<p> \tarithmetic sequence<br \/>\n \ta sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence<\/p>\n<dl id=\"fs-id1169739249458\">\n<dt>bounded above<\/dt>\n<dd id=\"fs-id1169739249464\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded above if there exists a constant [latex]M[\/latex] such that [latex]{a}_{n}\\le M[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739249504\">\n<dt>bounded below<\/dt>\n<dd id=\"fs-id1169739249509\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded below if there exists a constant [latex]M[\/latex] such that [latex]M\\le {a}_{n}[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739249549\">\n<dt>bounded sequence<\/dt>\n<dd id=\"fs-id1169739249554\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is bounded if there exists a constant [latex]M[\/latex] such that [latex]|{a}_{n}|\\le M[\/latex] for all positive integers [latex]n[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739249599\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt>comparison test<\/dt>\n<dd id=\"fs-id1169736778033\">if [latex]0\\le {a}_{n}\\le {b}_{n}[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges; if [latex]{a}_{n}\\ge {b}_{n}\\ge 0[\/latex] for all [latex]n\\ge N[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738040638\">\n<dt>conditional convergence<\/dt>\n<dd id=\"fs-id1169738040642\">if the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges, but the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }|{a}_{n}|[\/latex] diverges, the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] is said to converge conditionally<\/dd>\n<\/dl>\n<p> \tconvergence of a series<br \/>\n \ta series converges if the sequence of partial sums for that series converges<\/p>\n<p> \tconvergent sequence<br \/>\n \ta convergent sequence is a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] for which there exists a real number [latex]L[\/latex] such that [latex]{a}_{n}[\/latex] is arbitrarily close to [latex]L[\/latex] as long as [latex]n[\/latex] is sufficiently large<\/p>\n<dl>\n<dt>\n<\/dt>\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-id1169738154867\">\n<dt>divergence test<\/dt>\n<dd id=\"fs-id1169738154872\">if [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}\\ne 0[\/latex], then the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739249646\">\n<dt>divergent sequence<\/dt>\n<dd id=\"fs-id1169736702608\">a sequence that is not convergent is divergent<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736702613\">\n<dt>explicit formula<\/dt>\n<dd id=\"fs-id1169736702618\">a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\\left(n\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736702642\">\n<dt>geometric sequence<\/dt>\n<dd id=\"fs-id1169736702647\">a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] in which the ratio [latex]\\frac{{a}_{n+1}}{{a}_{n}}[\/latex] is the same for all positive integers [latex]n[\/latex] is called a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736702692\">\n<dt>\n<\/dt>\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<p> \tindex variable<br \/>\n \tthe subscript used to define the terms in a sequence is called the index<\/p>\n<dl id=\"fs-id1169736702702\">\n<dt>\n<\/dt>\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-id1169738154932\">\n<dt>integral test<\/dt>\n<dd id=\"fs-id1169738154937\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex], if there exists a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], then<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738161782\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736592441\">\n<dt>limit comparison test<\/dt>\n<dd id=\"fs-id1169736592446\">suppose [latex]{a}_{n},{b}_{n}\\ge 0[\/latex] for all [latex]n\\ge 1[\/latex]. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to L\\ne 0[\/latex], then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] both converge or both diverge; if [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to 0[\/latex] and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] converges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] converges. If [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{{a}_{n}}{{b}_{n}}\\to \\infty[\/latex], and [latex]\\displaystyle\\sum _{n=1}^{\\infty }{b}_{n}[\/latex] diverges, then [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] diverges<\/dd>\n<\/dl>\n<p> \tlimit of a sequence<br \/>\n \tthe real number [latex]L[\/latex] to which a sequence converges is called the limit of the sequence<\/p>\n<dl id=\"fs-id1169736702717\">\n<dt>monotone sequence<\/dt>\n<dd id=\"fs-id1169736702722\">an increasing or decreasing sequence<\/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-id1169737433691\">\n<dt><em data-effect=\"italics\">p<\/em>-series<\/dt>\n<dd id=\"fs-id1169737433701\">a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\n<\/dl>\n<p> \tratio test<br \/>\n \tfor a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with nonzero terms, let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}|\\frac{{a}_{n+1}}{{a}_{n}}|[\/latex]; if [latex]0\\le \\rho <1[\/latex], the series converges absolutely; if [latex]\\rho >1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/p>\n<dl id=\"fs-id1169736845132\">\n<dt>\n<\/dt>\n<dt>recurrence relation<\/dt>\n<dd id=\"fs-id1169736702732\">a recurrence relation is a relationship in which a term [latex]{a}_{n}[\/latex] in a sequence is defined in terms of earlier terms in the sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1169737174578\">\n<dt>remainder estimate<\/dt>\n<dd id=\"fs-id1169737174583\">for a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex] with positive terms [latex]{a}_{n}[\/latex] and a continuous, decreasing function [latex]f[\/latex] such that [latex]f\\left(n\\right)={a}_{n}[\/latex] for all positive integers [latex]n[\/latex], the remainder [latex]{R}_{N}=\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}-\\displaystyle\\sum _{n=1}^{N}{a}_{n}[\/latex] satisfies the following estimate:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169738066622\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<p> \troot test<br \/>\n \tfor a series [latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex], let [latex]\\rho =\\underset{n\\to \\infty }{\\text{lim}}\\sqrt[n]{|{a}_{n}|}[\/latex]; if [latex]0\\le \\rho <1[\/latex], the series converges absolutely; if [latex]\\rho >1[\/latex], the series diverges; if [latex]\\rho =1[\/latex], the test is inconclusive<\/p>\n<dl id=\"fs-id1169736702745\">\n<dt>sequence<\/dt>\n<dd id=\"fs-id1169736702750\">an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\\text{,}\\ldots[\/latex] is a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736702783\">\n<dt>\n<\/dt>\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<p> \tterm<br \/>\n \tthe number [latex]{a}_{n}[\/latex] in the sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] is called the [latex]n\\text{th}[\/latex] term of the sequence<\/p>\n<dl id=\"fs-id1169736634909\">\n<dt>unbounded sequence<\/dt>\n<dd id=\"fs-id1169736634914\">a sequence that is not bounded is called unbounded<\/dd>\n<\/dl>\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":671,"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\/849"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/849\/revisions"}],"predecessor-version":[{"id":1619,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/849\/revisions\/1619"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/671"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/849\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=849"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=849"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=849"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}