{"id":1857,"date":"2023-10-12T00:32:21","date_gmt":"2023-10-12T00:32:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-series-and-their-notations\/"},"modified":"2023-10-12T00:32:21","modified_gmt":"2023-10-12T00:32:21","slug":"introduction-series-and-their-notations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-series-and-their-notations\/","title":{"raw":"Series and Their Notations","rendered":"Series and Their Notations"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n \t<li class=\"li2\"><span class=\"s1\">Use summation notation.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of a series.<\/span><\/li>\n<\/ul>\n<\/div>\n<h3>Using Summation Notation<\/h3>\nTo find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a <strong>series<\/strong>. Consider, for example, the following series.\n<p style=\"text-align: center;\">[latex]3+7+11+15+19+\\cdots[\/latex]<\/p>\nThe <strong>[latex]n\\text{th }[\/latex] partial sum<\/strong> of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{S}_{n}\\text{ represents the partial sum.} \\\\ &amp;{S}_{1}=3 \\\\ &amp;{S}_{2}=3+7=10 \\\\ &amp;{S}_{3}=3+7+11=21 \\\\ &amp;{S}_{4}=3+7+11+15=36\\end{align}[\/latex]<\/p>\n<strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>, [latex]\\Sigma[\/latex], to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03225157\/CNX_Precalc_Figure_11_04_001n2.jpg\" alt=\"Explanation of summation notion as described in the text.\">\n\nIf we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5[\/latex]. We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{a}_{1}=2\\left(1\\right)=2 \\\\ &amp;{a}_{2}=2\\left(2\\right)=4 \\\\ &amp;{a}_{3}=2\\left(3\\right)=6 \\\\ &amp;{a}_{4}=2\\left(4\\right)=8 \\\\ &amp;{a}_{5}=2\\left(5\\right)=10 \\end{align}[\/latex]<\/p>\nWe can find the sum of the series by adding the terms:\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Summation Notation<\/h3>\nThe sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{n}{a}_{k}[\/latex]<\/p>\nThis notation tells us to find the sum of [latex]{a}_{k}[\/latex] from\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex].<\/p>\n[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Does the lower limit of summation have to be 1?<\/h4>\n<em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given summation notation for a series, evaluate the value.<\/h3>\n<ol>\n \t<li>Identify the lower limit of summation.<\/li>\n \t<li>Identify the upper limit of summation.<\/li>\n \t<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\n \t<li>Add to find the sum.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Summation Notation<\/h3>\nEvaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex].\n\n[reveal-answer q=\"991305\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"991305\"]\n\nAccording to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex]. We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex]. We add the terms to find the sum.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{k=3}^{7}{k}^{2} &amp; ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2} \\\\ &amp; =9+16+25+36+49 \\\\ &amp; =135 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nEvaluate [latex]\\sum\\limits _{k=2}^{5}\\left(3k - 1\\right)[\/latex].\n\n[reveal-answer q=\"788016\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"788016\"]\n\n38\n\n[\/hidden-answer]\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5866&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n<div class=\"textbox key-takeaways\">\n\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=128791&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\n\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The sum of the terms in a sequence is called a series.<\/li>\n \t<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>index of summation<\/strong> in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation\n\n<strong>lower limit of summation<\/strong> the number used in the explicit formula to find the first term in a series\n\n<strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series\n\n<strong>upper limit of summation<\/strong> the number used in the explicit formula to find the last term in a series\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Use summation notation.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of a series.<\/span><\/li>\n<\/ul>\n<\/div>\n<h3>Using Summation Notation<\/h3>\n<p>To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a <strong>series<\/strong>. Consider, for example, the following series.<\/p>\n<p style=\"text-align: center;\">[latex]3+7+11+15+19+\\cdots[\/latex]<\/p>\n<p>The <strong>[latex]n\\text{th }[\/latex] partial sum<\/strong> of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{S}_{n}\\text{ represents the partial sum.} \\\\ &{S}_{1}=3 \\\\ &{S}_{2}=3+7=10 \\\\ &{S}_{3}=3+7+11=21 \\\\ &{S}_{4}=3+7+11+15=36\\end{align}[\/latex]<\/p>\n<p><strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>, [latex]\\Sigma[\/latex], to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03225157\/CNX_Precalc_Figure_11_04_001n2.jpg\" alt=\"Explanation of summation notion as described in the text.\" \/><\/p>\n<p>If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5[\/latex]. We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{a}_{1}=2\\left(1\\right)=2 \\\\ &{a}_{2}=2\\left(2\\right)=4 \\\\ &{a}_{3}=2\\left(3\\right)=6 \\\\ &{a}_{4}=2\\left(4\\right)=8 \\\\ &{a}_{5}=2\\left(5\\right)=10 \\end{align}[\/latex]<\/p>\n<p>We can find the sum of the series by adding the terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Summation Notation<\/h3>\n<p>The sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{n}{a}_{k}[\/latex]<\/p>\n<p>This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from<\/p>\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex].<\/p>\n<p>[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Does the lower limit of summation have to be 1?<\/h4>\n<p><em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given summation notation for a series, evaluate the value.<\/h3>\n<ol>\n<li>Identify the lower limit of summation.<\/li>\n<li>Identify the upper limit of summation.<\/li>\n<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\n<li>Add to find the sum.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Summation Notation<\/h3>\n<p>Evaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q991305\">Show Solution<\/span><\/p>\n<div id=\"q991305\" class=\"hidden-answer\" style=\"display: none\">\n<p>According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex]. We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex]. We add the terms to find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{k=3}^{7}{k}^{2} & ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2} \\\\ & =9+16+25+36+49 \\\\ & =135 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\sum\\limits _{k=2}^{5}\\left(3k - 1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788016\">Show Solution<\/span><\/p>\n<div id=\"q788016\" class=\"hidden-answer\" style=\"display: none\">\n<p>38<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5866&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=128791&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The sum of the terms in a sequence is called a series.<\/li>\n<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>index of summation<\/strong> in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation<\/p>\n<p><strong>lower limit of summation<\/strong> the number used in the explicit formula to find the first term in a series<\/p>\n<p><strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/p>\n<p><strong>upper limit of summation<\/strong> the number used in the explicit formula to find the last term in a series<\/p>\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-1857\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 5865, 5867. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>:  IMathAS Community License CC-BY +GPL<\/li><li>Question ID 128790,128791. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 20277. <strong>Authored by<\/strong>: Kissel, Kris. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 23741. <strong>Authored by<\/strong>: Shahbazian, Roy, mb McClure, Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":708740,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 5865, 5867\",\"author\":\"WebWork-Rochester\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\" IMathAS Community License CC-BY +GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 128790,128791\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 20277\",\"author\":\"Kissel, Kris\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 23741\",\"author\":\"Shahbazian, Roy, mb McClure, Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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