{"id":693,"date":"2021-05-10T19:04:10","date_gmt":"2021-05-10T19:04:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=693"},"modified":"2021-11-17T03:00:10","modified_gmt":"2021-11-17T03:00:10","slug":"summary-of-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-sequences\/","title":{"raw":"Summary of Sequences","rendered":"Summary of Sequences"},"content":{"raw":"<section id=\"fs-id1169736857064\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\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<\/div>\r\n<\/section><section id=\"fs-id1169739236347\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739249448\">\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>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 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>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>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<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<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>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>\r\n<\/div>","rendered":"<section id=\"fs-id1169736857064\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\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<\/div>\n<\/section>\n<section id=\"fs-id1169739236347\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739249448\">\n<dt>arithmetic sequence<\/dt>\n<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>\n<\/dl>\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>convergent sequence<\/dt>\n<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>\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>index variable<\/dt>\n<dd id=\"fs-id1169736702698\">the subscript used to define the terms in a sequence is called the index<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736702702\">\n<dt>limit of a sequence<\/dt>\n<dd id=\"fs-id1169736702707\">the real number [latex]L[\/latex] to which a sequence converges is called the limit of the sequence<\/dd>\n<\/dl>\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-id1169736702727\">\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-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>term<\/dt>\n<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>\n<\/dl>\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<\/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-693\">\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":6,"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 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