{"id":1490,"date":"2025-07-25T02:11:06","date_gmt":"2025-07-25T02:11:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1490"},"modified":"2026-03-24T07:10:48","modified_gmt":"2026-03-24T07:10:48","slug":"series-and-their-notations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-fresh-take\/","title":{"raw":"Series and Their Notations: Fresh Take","rendered":"Series and Their Notations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use summation notation to represent a series.<\/li>\r\n \t<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\r\n \t<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\r\n \t<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\r\n \t<li>Solve word problems involving series.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Series and Summation Notation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Series Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A series is the sum of the terms in a sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">It represents the total of a sequence of numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Summation Notation (Sigma Notation):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Uses the Greek capital letter sigma ([latex]\\Sigma[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]\\sum_{k=1}^{n} a_k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represents the sum of [latex]a_k[\/latex] from [latex]k=1[\/latex] to [latex]k=n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components of Summation Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Index of summation: The variable used (e.g., [latex]k[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lower limit of summation: The starting value for the index<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Upper limit of summation: The ending value for the index<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Explicit formula: The expression to the right of [latex]\\Sigma[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluating a Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute each value of the index from lower to upper limit into the formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add all resulting terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\sum\\limits _{k=2}^{5}\\left(3k - 1\\right)[\/latex].[reveal-answer q=\"788016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"788016\"][latex]38[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eebeffcb-0L0rU17hHuM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0L0rU17hHuM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-eebeffcb-0L0rU17hHuM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851339&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eebeffcb-0L0rU17hHuM&amp;vembed=0&amp;video_id=0L0rU17hHuM&amp;video_target=tpm-plugin-eebeffcb-0L0rU17hHuM\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+a+Sum+Written+in+Summation+%3A+Sigma+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find a Sum Written in Summation \/ Sigma Notation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Arithmetic Series<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Arithmetic Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Sum of terms in an arithmetic sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Arithmetic sequence: difference between consecutive terms is constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula for Partial Sum: [latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]S_n[\/latex]: Sum of first [latex]n[\/latex] terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: Number of terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: [latex]n[\/latex]th term<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding [latex]a_n[\/latex] in Arithmetic Sequence: [latex]a_n = a_1 + (n - 1)d[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]d[\/latex]: Common difference<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial sum of each arithmetic series.\r\n<ol>\r\n \t<li>[latex]5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32[\/latex]<\/li>\r\n \t<li>[latex]20 + 15 + 10 + \\dots + -50[\/latex]<\/li>\r\n \t<li>[latex]\\sum\\limits _{k=1}^{12}3k - 8[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"470866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"470866\"]\r\n<ol>\r\n \t<li>We are given [latex]{a}_{1}=5[\/latex] and [latex]{a}_{n}=32[\/latex].Count the number of terms in the sequence to find [latex]n=10[\/latex].Substitute values for [latex]{a}_{1},{a}_{n},[\/latex] and [latex]n[\/latex] into the formula and simplify.\r\n[latex]\\begin{align} \\\\ &amp;{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &amp;{S}_{10}=\\dfrac{10\\left(5+32\\right)}{2}=185 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].\r\n[latex]\\begin{align}\\\\ {a}_{n}&amp;={a}_{1}+\\left(n - 1\\right)d \\\\ -50&amp;=20+\\left(n - 1\\right)\\left(-5\\right) \\\\ -70&amp;=\\left(n - 1\\right)\\left(-5\\right) \\\\ 14&amp;=n - 1 \\\\ 15&amp;=n \\\\ \\text{ }\\end{align}[\/latex]\r\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.\r\n[latex]\\begin{align}\\\\ &amp;{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &amp;{S}_{15}=\\dfrac{15\\left(20 - 50\\right)}{2}=-225 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.\r\n[latex]\\begin{align}\\\\ {a}_{k}&amp;=3k - 8 \\\\ {a}_{1}&amp;=3\\left(1\\right)-8=-5 \\\\ \\text{ }\\end{align}[\/latex]\r\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.\r\n[latex]\\begin{align} \\\\{a}_{k}&amp;=3k - 8 \\\\ {a}_{12}&amp;=3\\left(12\\right)-8=28 \\\\ \\text{ }\\end{align}[\/latex]\r\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.\r\n[latex]\\begin{align}\\\\{S}_{n}&amp;=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{12}&amp;=\\dfrac{12\\left(-5+28\\right)}{2}=138 \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the partial sum of each arithmetic series.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\u00a0[latex]1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4[\/latex][reveal-answer q=\"649728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"649728\"][latex]26.4[\/latex][\/hidden-answer]<\/li>\r\n \t<li>\u00a0[latex]12+21+29\\dots + 69[\/latex][reveal-answer q=\"617640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617640\"][latex]328[\/latex][\/hidden-answer]<\/li>\r\n \t<li>[latex]\\sum\\limits _{k=1}^{10}5 - 6k[\/latex][reveal-answer q=\"794771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"794771\"][latex]-280[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcdchceb-GZH68SubgRE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/GZH68SubgRE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcdchceb-GZH68SubgRE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851340&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gcdchceb-GZH68SubgRE&amp;vembed=0&amp;video_id=GZH68SubgRE&amp;video_target=tpm-plugin-gcdchceb-GZH68SubgRE\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Arithmetic+Series_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Arithmetic Series\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Geometric Series<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Geometric Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Sum of terms in a geometric sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric sequence: ratio between consecutive terms is constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula for Sum of First n Terms: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}, \\text{ where } r \\neq 1[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]S_n[\/latex]: Sum of first [latex]n[\/latex] terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: Number of terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the indicated partial sum of each geometric series.\r\n[latex]{S}_{20}[\/latex] for the series [latex]1\\text{,}000 + 500 + 250 + \\dots [\/latex][reveal-answer q=\"922435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"922435\"][latex]\\approx 2,000.00[\/latex][\/hidden-answer]Use the formula to determine the sum\u00a0[latex]\\sum\\limits _{k=1}^{8}{3}^{k}[\/latex][reveal-answer q=\"15208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15208\"][latex]9,840[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ghgebdef-mYg5gKlJjHc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mYg5gKlJjHc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ghgebdef-mYg5gKlJjHc\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851341&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghgebdef-mYg5gKlJjHc&amp;vembed=0&amp;video_id=mYg5gKlJjHc&amp;video_target=tpm-plugin-ghgebdef-mYg5gKlJjHc\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Geometric+Series_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGeometric Series\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Infinite Geometric Series<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Infinite Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Sum of the terms of an infinite sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]2 + 4 + 6 + 8 + \\cdots = \\sum_{k=1}^{\\infty} 2k[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convergence and Divergence:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Convergent: Sum approaches a finite value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divergent: Sum is not defined (increases without bound)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convergence of Infinite Geometric Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Converges when [latex]-1 &lt; r &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio of the geometric sequence<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula for Sum of Infinite Geometric Series: [latex]S_{\\infty} = \\frac{a_1}{1-r}, \\text{ where } |r| &lt; 1[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]S_{\\infty}[\/latex]: Sum of infinite terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the sum of the infinite series is defined.\r\n<ol>\r\n \t<li>[latex]\\dfrac{1}{3}+\\dfrac{1}{2}+\\dfrac{3}{4}+\\dfrac{9}{8}+\\cdots[\/latex]<\/li>\r\n \t<li>[latex]24+(-12)+6+(-3)+\\dots[\/latex]<\/li>\r\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty} 15\\cdot(-0.3)^k[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"559520\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"559520\"]\r\n<ol>\r\n \t<li>The series is geometric, but [latex]r=\\dfrac{3}{2}&gt;1[\/latex]. The sum is not defined.<\/li>\r\n \t<li>The series is geometric with [latex]r=-\\dfrac{1}{2}[\/latex]. The sum is defined.<\/li>\r\n \t<li>The series is geometric with [latex]r=-0.3[\/latex]. The sum is defined.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Finding Sums of Infinite Series<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Formula for Sum of Infinite Geometric Series: [latex]S = \\frac{a_1}{1-r}, \\text{ where } |r| &lt; 1[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]S[\/latex]: Sum of infinite terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Derivation from Finite Series:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Finite series sum: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]n \\to \\infty[\/latex], [latex]r^n \\to 0[\/latex] when [latex]|r| &lt; 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convergence Condition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Series converges when [latex]-1 &lt; r &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r^n[\/latex] approaches [latex]0[\/latex] as [latex]n[\/latex] increases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Converting repeating decimals to fractions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving real-world problems involving infinite processes<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum if it exists.\r\n<ol>\r\n \t<li>[latex]2+\\dfrac{2}{3}+\\dfrac{2}{9}+\\dots[\/latex]<\/li>\r\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}{0.76k+1}[\/latex]<\/li>\r\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\left(-\\dfrac{3}{8}\\right)^k[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"221023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221023\"]\r\n<ol>\r\n \t<li>\u00a03<\/li>\r\n \t<li>\u00a0The series is arithmetic. The sum does not exist.<\/li>\r\n \t<li>\u00a0[latex]-\\dfrac{3}{11}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use summation notation to represent a series.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\n<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\n<li>Solve word problems involving series.<\/li>\n<\/ul>\n<\/section>\n<h2>Series and Summation Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Series Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A series is the sum of the terms in a sequence<\/li>\n<li class=\"whitespace-normal break-words\">It represents the total of a sequence of numbers<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Summation Notation (Sigma Notation):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Uses the Greek capital letter sigma ([latex]\\Sigma[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]\\sum_{k=1}^{n} a_k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represents the sum of [latex]a_k[\/latex] from [latex]k=1[\/latex] to [latex]k=n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components of Summation Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Index of summation: The variable used (e.g., [latex]k[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Lower limit of summation: The starting value for the index<\/li>\n<li class=\"whitespace-normal break-words\">Upper limit of summation: The ending value for the index<\/li>\n<li class=\"whitespace-normal break-words\">Explicit formula: The expression to the right of [latex]\\Sigma[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating a Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute each value of the index from lower to upper limit into the formula<\/li>\n<li class=\"whitespace-normal break-words\">Add all resulting terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\sum\\limits _{k=2}^{5}\\left(3k - 1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q788016\">Show Solution<\/button><\/p>\n<div id=\"q788016\" class=\"hidden-answer\" style=\"display: none\">[latex]38[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eebeffcb-0L0rU17hHuM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0L0rU17hHuM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-eebeffcb-0L0rU17hHuM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851339&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-eebeffcb-0L0rU17hHuM&amp;vembed=0&amp;video_id=0L0rU17hHuM&amp;video_target=tpm-plugin-eebeffcb-0L0rU17hHuM\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+a+Sum+Written+in+Summation+%3A+Sigma+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find a Sum Written in Summation \/ Sigma Notation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Arithmetic Series<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Arithmetic Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum of terms in an arithmetic sequence<\/li>\n<li class=\"whitespace-normal break-words\">Arithmetic sequence: difference between consecutive terms is constant<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula for Partial Sum: [latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]S_n[\/latex]: Sum of first [latex]n[\/latex] terms<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: Number of terms<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: [latex]n[\/latex]th term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding [latex]a_n[\/latex] in Arithmetic Sequence: [latex]a_n = a_1 + (n - 1)d[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]d[\/latex]: Common difference<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the partial sum of each arithmetic series.<\/p>\n<ol>\n<li>[latex]5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32[\/latex]<\/li>\n<li>[latex]20 + 15 + 10 + \\dots + -50[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{12}3k - 8[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q470866\">Show Solution<\/button><\/p>\n<div id=\"q470866\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We are given [latex]{a}_{1}=5[\/latex] and [latex]{a}_{n}=32[\/latex].Count the number of terms in the sequence to find [latex]n=10[\/latex].Substitute values for [latex]{a}_{1},{a}_{n},[\/latex] and [latex]n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align} \\\\ &{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &{S}_{10}=\\dfrac{10\\left(5+32\\right)}{2}=185 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].<br \/>\n[latex]\\begin{align}\\\\ {a}_{n}&={a}_{1}+\\left(n - 1\\right)d \\\\ -50&=20+\\left(n - 1\\right)\\left(-5\\right) \\\\ -70&=\\left(n - 1\\right)\\left(-5\\right) \\\\ 14&=n - 1 \\\\ 15&=n \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align}\\\\ &{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &{S}_{15}=\\dfrac{15\\left(20 - 50\\right)}{2}=-225 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{align}\\\\ {a}_{k}&=3k - 8 \\\\ {a}_{1}&=3\\left(1\\right)-8=-5 \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{align} \\\\{a}_{k}&=3k - 8 \\\\ {a}_{12}&=3\\left(12\\right)-8=28 \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align}\\\\{S}_{n}&=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{12}&=\\dfrac{12\\left(-5+28\\right)}{2}=138 \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the partial sum of each arithmetic series.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\u00a0[latex]1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q649728\">Show Solution<\/button><\/p>\n<div id=\"q649728\" class=\"hidden-answer\" style=\"display: none\">[latex]26.4[\/latex]<\/div>\n<\/div>\n<\/li>\n<li>\u00a0[latex]12+21+29\\dots + 69[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q617640\">Show Solution<\/button><\/p>\n<div id=\"q617640\" class=\"hidden-answer\" style=\"display: none\">[latex]328[\/latex]<\/div>\n<\/div>\n<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{10}5 - 6k[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q794771\">Show Solution<\/button><\/p>\n<div id=\"q794771\" class=\"hidden-answer\" style=\"display: none\">[latex]-280[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcdchceb-GZH68SubgRE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/GZH68SubgRE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcdchceb-GZH68SubgRE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851340&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gcdchceb-GZH68SubgRE&amp;vembed=0&amp;video_id=GZH68SubgRE&amp;video_target=tpm-plugin-gcdchceb-GZH68SubgRE\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Arithmetic+Series_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Arithmetic Series\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Geometric Series<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Geometric Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum of terms in a geometric sequence<\/li>\n<li class=\"whitespace-normal break-words\">Geometric sequence: ratio between consecutive terms is constant<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula for Sum of First n Terms: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}, \\text{ where } r \\neq 1[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]S_n[\/latex]: Sum of first [latex]n[\/latex] terms<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: Number of terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the indicated partial sum of each geometric series.<br \/>\n[latex]{S}_{20}[\/latex] for the series [latex]1\\text{,}000 + 500 + 250 + \\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q922435\">Show Solution<\/button><\/p>\n<div id=\"q922435\" class=\"hidden-answer\" style=\"display: none\">[latex]\\approx 2,000.00[\/latex]<\/div>\n<\/div>\n<p>Use the formula to determine the sum\u00a0[latex]\\sum\\limits _{k=1}^{8}{3}^{k}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q15208\">Show Solution<\/button><\/p>\n<div id=\"q15208\" class=\"hidden-answer\" style=\"display: none\">[latex]9,840[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ghgebdef-mYg5gKlJjHc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mYg5gKlJjHc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ghgebdef-mYg5gKlJjHc\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851341&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghgebdef-mYg5gKlJjHc&amp;vembed=0&amp;video_id=mYg5gKlJjHc&amp;video_target=tpm-plugin-ghgebdef-mYg5gKlJjHc\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Geometric+Series_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGeometric Series\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Infinite Geometric Series<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Infinite Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum of the terms of an infinite sequence<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]2 + 4 + 6 + 8 + \\cdots = \\sum_{k=1}^{\\infty} 2k[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Convergence and Divergence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Convergent: Sum approaches a finite value<\/li>\n<li class=\"whitespace-normal break-words\">Divergent: Sum is not defined (increases without bound)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Convergence of Infinite Geometric Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Converges when [latex]-1 < r < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio of the geometric sequence<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula for Sum of Infinite Geometric Series: [latex]S_{\\infty} = \\frac{a_1}{1-r}, \\text{ where } |r| < 1[\/latex]\n\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]S_{\\infty}[\/latex]: Sum of infinite terms<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the sum of the infinite series is defined.<\/p>\n<ol>\n<li>[latex]\\dfrac{1}{3}+\\dfrac{1}{2}+\\dfrac{3}{4}+\\dfrac{9}{8}+\\cdots[\/latex]<\/li>\n<li>[latex]24+(-12)+6+(-3)+\\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty} 15\\cdot(-0.3)^k[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q559520\">Show Solution<\/button><\/p>\n<div id=\"q559520\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The series is geometric, but [latex]r=\\dfrac{3}{2}>1[\/latex]. The sum is not defined.<\/li>\n<li>The series is geometric with [latex]r=-\\dfrac{1}{2}[\/latex]. The sum is defined.<\/li>\n<li>The series is geometric with [latex]r=-0.3[\/latex]. The sum is defined.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding Sums of Infinite Series<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Formula for Sum of Infinite Geometric Series: [latex]S = \\frac{a_1}{1-r}, \\text{ where } |r| < 1[\/latex]\n\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]S[\/latex]: Sum of infinite terms<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: First term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: Common ratio<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation from Finite Series:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Finite series sum: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]n \\to \\infty[\/latex], [latex]r^n \\to 0[\/latex] when [latex]|r| < 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Convergence Condition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Series converges when [latex]-1 < r < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r^n[\/latex] approaches [latex]0[\/latex] as [latex]n[\/latex] increases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Converting repeating decimals to fractions<\/li>\n<li class=\"whitespace-normal break-words\">Solving real-world problems involving infinite processes<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum if it exists.<\/p>\n<ol>\n<li>[latex]2+\\dfrac{2}{3}+\\dfrac{2}{9}+\\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}{0.76k+1}[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\left(-\\dfrac{3}{8}\\right)^k[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q221023\">Show Solution<\/button><\/p>\n<div id=\"q221023\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a03<\/li>\n<li>\u00a0The series is arithmetic. The sum does not exist.<\/li>\n<li>\u00a0[latex]-\\dfrac{3}{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Find a Sum Written in Summation \/ Sigma Notation\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/0L0rU17hHuM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Arithmetic Series\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/GZH68SubgRE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Geometric Series\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/mYg5gKlJjHc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard 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