{"id":160,"date":"2025-02-13T22:44:28","date_gmt":"2025-02-13T22:44:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-2\/"},"modified":"2026-03-26T15:59:08","modified_gmt":"2026-03-26T15:59:08","slug":"series-and-their-notations-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-2\/","title":{"raw":"Series and Their Notations: Learn It 1","rendered":"Series and Their Notations: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Use summation notation to represent a series.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve word problems involving series.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Series and Summation Notation<\/h2>\r\nNow that we've explored sequences, let's take it a step further and talk about series. While a sequence lists numbers in a specific order, a series is what you get when you add those numbers together. In other words, a series is the sum of the terms in a sequence. Understanding how to work with series is important because it allows us to find the total of a sequence of numbers, which has many practical applications in mathematics and beyond.\r\n\r\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.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\"><img class=\"aligncenter wp-image-4983\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-300x43.png\" alt=\"Explanation of summation notion as described in the text.\" width=\"579\" height=\"83\" \/>\r\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.\r\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>\r\nWe can find the sum of the series by adding the terms:\r\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>summation notation<\/h3>\r\nThe sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:\r\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{n}{a}_{k}[\/latex]<\/p>\r\nThis notation tells us to find the sum of [latex]{a}_{k}[\/latex] from\r\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex].<\/p>\r\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>.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given summation notation for a series, evaluate the value.<\/strong>\r\n<ol>\r\n \t<li>Identify the lower limit of summation.<\/li>\r\n \t<li>Identify the upper limit of summation.<\/li>\r\n \t<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\r\n \t<li>Add to find the sum.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use summation notation to write the sum:<center>[latex]1+6+11+16+21+26+31[\/latex]<\/center>[reveal-answer q=\"947803\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"947803\"]Note that [latex]1+6+11+16+21+26+31[\/latex] is the sum of the first seven terms of the arithmetic sequence <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">with general term [latex]a_n = 5n-4[\/latex].\r\n[latex]\\\\[\/latex]\r\n<\/span>This means that the summation notation of the series is\r\n<p style=\"text-align: center;\">[latex]\\sum_{n=1}^{7} (5n - 4)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex].[reveal-answer q=\"991305\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"991305\"]According to the notation, the lower limit of summation is [latex]3[\/latex] and the upper limit is [latex]7[\/latex]. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex].\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span>\r\nWe find the terms of the series by substituting [latex]k=3, 4, 5, 6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex].\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span>\r\nWe add the terms to find the sum.\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321978[\/ohm_question]<\/section><\/div>\r\n<dl id=\"fs-id1165137762744\" class=\"definition\">\r\n \t<dd id=\"fs-id1165137762749\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Use summation notation to represent a series.<\/li>\n<li style=\"font-weight: 400;\">Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\n<li style=\"font-weight: 400;\">Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\n<li style=\"font-weight: 400;\">Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\n<li style=\"font-weight: 400;\">Solve word problems involving series.<\/li>\n<\/ul>\n<\/section>\n<h2>Series and Summation Notation<\/h2>\n<p>Now that we&#8217;ve explored sequences, let&#8217;s take it a step further and talk about series. While a sequence lists numbers in a specific order, a series is what you get when you add those numbers together. In other words, a series is the sum of the terms in a sequence. Understanding how to work with series is important because it allows us to find the total of a sequence of numbers, which has many practical applications in mathematics and beyond.<\/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<section class=\"textbox example\" aria-label=\"Example\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4983\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-300x43.png\" alt=\"Explanation of summation notion as described in the text.\" width=\"579\" height=\"83\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-300x43.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-65x9.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-225x32.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram-350x50.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03000231\/11.1.L1.Diagram.png 747w\" sizes=\"(max-width: 579px) 100vw, 579px\" \/><br \/>\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.<\/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<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>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<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given summation notation for a series, evaluate the value.<\/strong><\/p>\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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use summation notation to write the sum:<\/p>\n<div style=\"text-align: center;\">[latex]1+6+11+16+21+26+31[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q947803\">Show Answer<\/button><\/p>\n<div id=\"q947803\" class=\"hidden-answer\" style=\"display: none\">Note that [latex]1+6+11+16+21+26+31[\/latex] is the sum of the first seven terms of the arithmetic sequence <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">with general term [latex]a_n = 5n-4[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\n<\/span>This means that the summation notation of the series is<\/p>\n<p style=\"text-align: center;\">[latex]\\sum_{n=1}^{7} (5n - 4)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q991305\">Show Solution<\/button><\/p>\n<div id=\"q991305\" class=\"hidden-answer\" style=\"display: none\">According to the notation, the lower limit of summation is [latex]3[\/latex] and the upper limit is [latex]7[\/latex]. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex].<br \/>\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span><br \/>\nWe find the terms of the series by substituting [latex]k=3, 4, 5, 6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex].<br \/>\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span><br \/>\nWe 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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321978\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321978&theme=lumen&iframe_resize_id=ohm321978&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<dl id=\"fs-id1165137762744\" class=\"definition\">\n<dd id=\"fs-id1165137762749\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":20,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"learn_it","content_attributions":[{"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":""}],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/160"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/160\/revisions"}],"predecessor-version":[{"id":6036,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/160\/revisions\/6036"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}