{"id":1116,"date":"2021-06-30T17:01:55","date_gmt":"2021-06-30T17:01:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/sigma-summation-notation\/"},"modified":"2022-03-19T03:10:50","modified_gmt":"2022-03-19T03:10:50","slug":"sigma-summation-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/sigma-summation-notation\/","title":{"raw":"Sigma Notation","rendered":"Sigma Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use sigma (summation) notation to calculate sums and powers of integers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572611083\" class=\"bc-section section\">\r\n<p id=\"fs-id1170572444485\">As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called<strong> sigma notation<\/strong> (also known as summation notation). The Greek capital letter [latex]\\Sigma[\/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write<\/p>\r\n\r\n<div id=\"fs-id1170572139479\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572184284\">We could probably skip writing a couple of terms and write<\/p>\r\n\r\n<div id=\"fs-id1170572550692\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+\\cdots+19+20[\/latex],<\/div>\r\nwhich is better, but still cumbersome. With sigma notation, we write this sum as\r\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572132999\">which is much more compact.<\/p>\r\n<p id=\"fs-id1170571654708\">Typically, sigma notation is presented in the form<\/p>\r\n\r\n<div id=\"fs-id1170571602104\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572108202\">where [latex]a_i[\/latex] describes the terms to be added, and the [latex]i[\/latex] is called the <span class=\"no-emphasis\"><em>index<\/em><\/span>. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[\/latex] and ending with the value when [latex]i=n[\/latex]. For example, an expression like [latex]\\displaystyle\\sum_{i=2}^{7} s_i[\/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[\/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a <span class=\"no-emphasis\"><em>dummy variable<\/em><\/span>. We can use any letter we like for the index. Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\r\n<p id=\"fs-id1170572472220\">Let\u2019s try a couple of examples of using sigma notation.<\/p>\r\n\r\n<div id=\"fs-id1170572294328\" class=\"textbook exercises\">\r\n<h3>Example: Using Sigma Notation<\/h3>\r\n<ol id=\"fs-id1170571671542\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Write in sigma notation and evaluate the sum of terms [latex]3^i[\/latex] for [latex]i=1,2,3,4,5[\/latex].<\/li>\r\n \t<li>Write the sum in sigma notation:\r\n<div class=\"equation unnumbered\">[latex]1+\\frac{1}{4}+\\frac{1}{9}+\\frac{1}{16}+\\frac{1}{25}[\/latex].<\/div><\/li>\r\n<\/ol>\r\n<div id=\"fs-id1170572142219\" class=\"exercise\">[reveal-answer q=\"fs-id1170572346829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572346829\"]\r\n<ol id=\"fs-id1170572346829\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Write\r\n<div id=\"fs-id1170572296964\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{5} 3^i &amp; =3+3^2+3^3+3^4+3^5 \\\\ &amp; =363 \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>The denominator of each term is a perfect square. Using sigma notation, this sum can be written as [latex]\\displaystyle\\sum_{i=1}^{5} \\frac{1}{i^2}[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Using Sigma Notation.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qx-gvr8k8SY?controls=0&amp;start=22&amp;end=150&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.1ApproximatingAreas22to150_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.1 Approximating Areas\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170571671517\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite in sigma notation and evaluate the sum of terms [latex]2^i[\/latex]\u00a0for [latex]i=3,4,5,6[\/latex].\r\n<div>[reveal-answer q=\"525019\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"525019\"]Use the solving steps in the last example as a guide.[\/hidden-answer]<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1170572107370\" class=\"exercise\">[reveal-answer q=\"fs-id1170572133846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572133846\"]\r\n<p id=\"fs-id1170572133846\">[latex]\\displaystyle\\sum_{i=3}^{6} 2^i=2^3+2^4+2^5+2^6=120[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572108042\">The properties associated with the summation process are given in the following rule.<\/p>\r\n\r\n<div id=\"fs-id1170572230416\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Properties of Sigma Notation<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572204646\">Let [latex]a_1,a_2, \\cdots,a_n[\/latex] and [latex]b_1,b_2,\\cdots,b_n[\/latex] represent two sequences of terms and let [latex]c[\/latex] be a constant. The following properties hold for all positive integers [latex]n[\/latex] and for integers [latex]m[\/latex], with [latex]1\\le m\\le n[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170571780454\">\r\n \t<li>\r\n<div id=\"fs-id1170572294298\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571809640\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170572138153\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571678952\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170572629258\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572216060\" class=\"bc-section section\">\r\n<h3>Proof<\/h3>\r\n<p id=\"fs-id1170572453570\">We prove properties 2 and 3 here, and leave proof of the other properties to the Exercises.<\/p>\r\n<p id=\"fs-id1167794031899\">Property 2: We have<\/p>\r\n\r\n<div id=\"fs-id1170571656289\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{n} ca_i &amp; =ca_1+ca_2+ca_3+\\cdots+ca_n \\\\ &amp; =c(a_1+a_2+a_3+\\cdots+a_n) \\\\ &amp; =c\\displaystyle\\sum_{i=1}^{n} a_i \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1167793969731\">Property 3: We have<\/p>\r\n\r\n<div id=\"fs-id1170572103031\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{n} (a_i+b_i) &amp; =(a_1+b_1)+(a_2+b_2)+(a_3+b_3)+\\cdots+(a_n+b_n) \\\\ &amp; =(a_1+a_2+a_3+\\cdots+a_n)+(b_1+b_2+b_3+\\cdots+b_n) \\\\ &amp; =\\displaystyle\\sum_{i=1}^{n} a_i+ \\displaystyle\\sum_{i=1}^{n} b_i \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572110150\">[latex]_\\blacksquare[\/latex]<\/p>\r\n<p id=\"fs-id1170572451163\">A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers, and we use them in the next set of examples.<\/p>\r\n\r\n<div id=\"fs-id1170572250513\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Sums and Powers of Integers<\/h3>\r\n\r\n<hr \/>\r\n\r\n<ol id=\"fs-id1170572608710\">\r\n \t<li>The sum of [latex]n[\/latex] integers is given by\r\n<div id=\"fs-id1170572241371\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex].<\/div><\/li>\r\n \t<li>The sum of consecutive integers squared is given by\r\n<div id=\"fs-id1170572560041\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex].<\/div><\/li>\r\n \t<li>The sum of consecutive integers cubed is given by\r\n<div id=\"fs-id1170572093566\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex].<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572218770\" class=\"textbook exercises\">\r\n<h3>Example: Evaluation Using Sigma Notation<\/h3>\r\n<p id=\"fs-id1170572297312\">Write using sigma notation and evaluate:<\/p>\r\n\r\n<ol id=\"fs-id1170571635786\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>The sum of the terms [latex](i-3)^2[\/latex] for [latex]i=1,2,\\cdots,200[\/latex].<\/li>\r\n \t<li>The sum of the terms [latex](i^3-i^2)[\/latex] for [latex]i=1,2,3,4,5,6[\/latex].<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1170572370645\" class=\"exercise\">[reveal-answer q=\"fs-id1170572088097\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572088097\"]\r\n<ol id=\"fs-id1170572088097\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Multiplying out [latex](i-3)^2[\/latex], we can break the expression into three terms.\r\n<div id=\"fs-id1170572107292\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\underset{i=1}{\\overset{200}{\\Sigma}}(i-3)^2 &amp; =\\underset{i=1}{\\overset{200}{\\Sigma}}(i^2-6i+9) \\\\ &amp; =\\underset{i=1}{\\overset{200}{\\Sigma}}i^2-\\underset{i=1}{\\overset{200}{\\Sigma}}6i+\\underset{i=1}{\\overset{200}{\\Sigma}}9 \\\\ &amp; =\\underset{i=1}{\\overset{200}{\\Sigma}}i^2-6\\underset{i=1}{\\overset{200}{\\Sigma}}i+\\underset{i=1}{\\overset{200}{\\Sigma}}9 \\\\ &amp; =\\frac{200(200+1)(400+1)}{6}-6[\\frac{200(200+1)}{2}]+9(200) \\\\ &amp; =2,686,700-120,600+1800 \\\\ &amp; =2,567,900 \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.\r\n<div id=\"fs-id1170572140369\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\underset{i=1}{\\overset{6}{\\Sigma}}(i^3-i^2) &amp; =\\underset{i=1}{\\overset{6}{\\Sigma}}i^3-\\underset{i=1}{\\overset{6}{\\Sigma}}i^2 \\\\ &amp; =\\frac{6^2(6+1)^2}{4}-\\frac{6(6+1)(2(6)+1)}{6} \\\\ &amp; =\\frac{1764}{4}-\\frac{546}{6} \\\\ &amp; =350 \\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Evaluation Using Sigma Notation.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qx-gvr8k8SY?controls=0&amp;start=194&amp;end=377&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.1ApproximatingAreas194to377_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.1 Approximating Areas\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the sum of the values of [latex]4+3i[\/latex] for [latex]i=1,2,\\cdots,100[\/latex].\r\n\r\n[reveal-answer q=\"12359899\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"12359899\"]\r\n\r\nUse the properties of sigma notation to solve the problem.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572292992\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572292992\"]\r\n\r\n[latex]15,550[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572337091\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Sum of the Function Values<\/h3>\r\nFind the sum of the values of [latex]f(x)=x^3[\/latex] over the integers [latex]1,2,3,\\cdots,10[\/latex].\r\n<div id=\"fs-id1170572337093\" class=\"exercise\">[reveal-answer q=\"fs-id1170571599435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571599435\"]\r\n<p id=\"fs-id1170571599435\">Using the formula, we have<\/p>\r\n\r\n<div id=\"fs-id1170572224759\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=0}^{10} i^3 &amp; =\\frac{(10)^2(10+1)^2}{4} \\\\ &amp; =\\frac{100(121)}{4} \\\\ &amp; =3025 \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572178187\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nEvaluate the sum indicated by the notation [latex]\\displaystyle\\sum_{k=1}^{20} (2k+1)[\/latex].\r\n<div id=\"fs-id1170572178190\" class=\"exercise\">[reveal-answer q=\"8326610\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8326610\"]\r\n<p id=\"fs-id1170571618983\">Use the rule on sum and powers of integers.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572280444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572280444\"]\r\n<p id=\"fs-id1170572280444\">[latex]440[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]19441[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use sigma (summation) notation to calculate sums and powers of integers<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572611083\" class=\"bc-section section\">\n<p id=\"fs-id1170572444485\">As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called<strong> sigma notation<\/strong> (also known as summation notation). The Greek capital letter [latex]\\Sigma[\/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write<\/p>\n<div id=\"fs-id1170572139479\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572184284\">We could probably skip writing a couple of terms and write<\/p>\n<div id=\"fs-id1170572550692\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+\\cdots+19+20[\/latex],<\/div>\n<p>which is better, but still cumbersome. With sigma notation, we write this sum as<\/p>\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572132999\">which is much more compact.<\/p>\n<p id=\"fs-id1170571654708\">Typically, sigma notation is presented in the form<\/p>\n<div id=\"fs-id1170571602104\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572108202\">where [latex]a_i[\/latex] describes the terms to be added, and the [latex]i[\/latex] is called the <span class=\"no-emphasis\"><em>index<\/em><\/span>. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[\/latex] and ending with the value when [latex]i=n[\/latex]. For example, an expression like [latex]\\displaystyle\\sum_{i=2}^{7} s_i[\/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[\/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a <span class=\"no-emphasis\"><em>dummy variable<\/em><\/span>. We can use any letter we like for the index. Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\n<p id=\"fs-id1170572472220\">Let\u2019s try a couple of examples of using sigma notation.<\/p>\n<div id=\"fs-id1170572294328\" class=\"textbook exercises\">\n<h3>Example: Using Sigma Notation<\/h3>\n<ol id=\"fs-id1170571671542\" style=\"list-style-type: lower-alpha;\">\n<li>Write in sigma notation and evaluate the sum of terms [latex]3^i[\/latex] for [latex]i=1,2,3,4,5[\/latex].<\/li>\n<li>Write the sum in sigma notation:\n<div class=\"equation unnumbered\">[latex]1+\\frac{1}{4}+\\frac{1}{9}+\\frac{1}{16}+\\frac{1}{25}[\/latex].<\/div>\n<\/li>\n<\/ol>\n<div id=\"fs-id1170572142219\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572346829\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572346829\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572346829\" style=\"list-style-type: lower-alpha;\">\n<li>Write\n<div id=\"fs-id1170572296964\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{5} 3^i & =3+3^2+3^3+3^4+3^5 \\\\ & =363 \\end{array}[\/latex]<\/div>\n<\/li>\n<li>The denominator of each term is a perfect square. Using sigma notation, this sum can be written as [latex]\\displaystyle\\sum_{i=1}^{5} \\frac{1}{i^2}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Using Sigma Notation.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qx-gvr8k8SY?controls=0&amp;start=22&amp;end=150&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.1ApproximatingAreas22to150_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.1 Approximating Areas&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170571671517\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write in sigma notation and evaluate the sum of terms [latex]2^i[\/latex]\u00a0for [latex]i=3,4,5,6[\/latex].<\/p>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525019\">Hint<\/span><\/p>\n<div id=\"q525019\" class=\"hidden-answer\" style=\"display: none\">Use the solving steps in the last example as a guide.<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div id=\"fs-id1170572107370\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572133846\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572133846\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572133846\">[latex]\\displaystyle\\sum_{i=3}^{6} 2^i=2^3+2^4+2^5+2^6=120[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572108042\">The properties associated with the summation process are given in the following rule.<\/p>\n<div id=\"fs-id1170572230416\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Properties of Sigma Notation<\/h3>\n<hr \/>\n<p id=\"fs-id1170572204646\">Let [latex]a_1,a_2, \\cdots,a_n[\/latex] and [latex]b_1,b_2,\\cdots,b_n[\/latex] represent two sequences of terms and let [latex]c[\/latex] be a constant. The following properties hold for all positive integers [latex]n[\/latex] and for integers [latex]m[\/latex], with [latex]1\\le m\\le n[\/latex].<\/p>\n<ol id=\"fs-id1170571780454\">\n<li>\n<div id=\"fs-id1170572294298\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170571809640\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170572138153\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170571678952\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170572629258\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572216060\" class=\"bc-section section\">\n<h3>Proof<\/h3>\n<p id=\"fs-id1170572453570\">We prove properties 2 and 3 here, and leave proof of the other properties to the Exercises.<\/p>\n<p id=\"fs-id1167794031899\">Property 2: We have<\/p>\n<div id=\"fs-id1170571656289\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{n} ca_i & =ca_1+ca_2+ca_3+\\cdots+ca_n \\\\ & =c(a_1+a_2+a_3+\\cdots+a_n) \\\\ & =c\\displaystyle\\sum_{i=1}^{n} a_i \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793969731\">Property 3: We have<\/p>\n<div id=\"fs-id1170572103031\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=1}^{n} (a_i+b_i) & =(a_1+b_1)+(a_2+b_2)+(a_3+b_3)+\\cdots+(a_n+b_n) \\\\ & =(a_1+a_2+a_3+\\cdots+a_n)+(b_1+b_2+b_3+\\cdots+b_n) \\\\ & =\\displaystyle\\sum_{i=1}^{n} a_i+ \\displaystyle\\sum_{i=1}^{n} b_i \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572110150\">[latex]_\\blacksquare[\/latex]<\/p>\n<p id=\"fs-id1170572451163\">A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers, and we use them in the next set of examples.<\/p>\n<div id=\"fs-id1170572250513\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Sums and Powers of Integers<\/h3>\n<hr \/>\n<ol id=\"fs-id1170572608710\">\n<li>The sum of [latex]n[\/latex] integers is given by\n<div id=\"fs-id1170572241371\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex].<\/div>\n<\/li>\n<li>The sum of consecutive integers squared is given by\n<div id=\"fs-id1170572560041\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex].<\/div>\n<\/li>\n<li>The sum of consecutive integers cubed is given by\n<div id=\"fs-id1170572093566\" class=\"equation unnumbered\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572218770\" class=\"textbook exercises\">\n<h3>Example: Evaluation Using Sigma Notation<\/h3>\n<p id=\"fs-id1170572297312\">Write using sigma notation and evaluate:<\/p>\n<ol id=\"fs-id1170571635786\" style=\"list-style-type: lower-alpha;\">\n<li>The sum of the terms [latex](i-3)^2[\/latex] for [latex]i=1,2,\\cdots,200[\/latex].<\/li>\n<li>The sum of the terms [latex](i^3-i^2)[\/latex] for [latex]i=1,2,3,4,5,6[\/latex].<\/li>\n<\/ol>\n<div id=\"fs-id1170572370645\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572088097\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572088097\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572088097\" style=\"list-style-type: lower-alpha;\">\n<li>Multiplying out [latex](i-3)^2[\/latex], we can break the expression into three terms.\n<div id=\"fs-id1170572107292\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\underset{i=1}{\\overset{200}{\\Sigma}}(i-3)^2 & =\\underset{i=1}{\\overset{200}{\\Sigma}}(i^2-6i+9) \\\\ & =\\underset{i=1}{\\overset{200}{\\Sigma}}i^2-\\underset{i=1}{\\overset{200}{\\Sigma}}6i+\\underset{i=1}{\\overset{200}{\\Sigma}}9 \\\\ & =\\underset{i=1}{\\overset{200}{\\Sigma}}i^2-6\\underset{i=1}{\\overset{200}{\\Sigma}}i+\\underset{i=1}{\\overset{200}{\\Sigma}}9 \\\\ & =\\frac{200(200+1)(400+1)}{6}-6[\\frac{200(200+1)}{2}]+9(200) \\\\ & =2,686,700-120,600+1800 \\\\ & =2,567,900 \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.\n<div id=\"fs-id1170572140369\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\underset{i=1}{\\overset{6}{\\Sigma}}(i^3-i^2) & =\\underset{i=1}{\\overset{6}{\\Sigma}}i^3-\\underset{i=1}{\\overset{6}{\\Sigma}}i^2 \\\\ & =\\frac{6^2(6+1)^2}{4}-\\frac{6(6+1)(2(6)+1)}{6} \\\\ & =\\frac{1764}{4}-\\frac{546}{6} \\\\ & =350 \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Evaluation Using Sigma Notation.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qx-gvr8k8SY?controls=0&amp;start=194&amp;end=377&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.1ApproximatingAreas194to377_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.1 Approximating Areas&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the sum of the values of [latex]4+3i[\/latex] for [latex]i=1,2,\\cdots,100[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q12359899\">Hint<\/span><\/p>\n<div id=\"q12359899\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the properties of sigma notation to solve the problem.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572292992\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572292992\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]15,550[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572337091\" class=\"textbook exercises\">\n<h3>Example: Finding the Sum of the Function Values<\/h3>\n<p>Find the sum of the values of [latex]f(x)=x^3[\/latex] over the integers [latex]1,2,3,\\cdots,10[\/latex].<\/p>\n<div id=\"fs-id1170572337093\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571599435\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571599435\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571599435\">Using the formula, we have<\/p>\n<div id=\"fs-id1170572224759\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}\\displaystyle\\sum_{i=0}^{10} i^3 & =\\frac{(10)^2(10+1)^2}{4} \\\\ & =\\frac{100(121)}{4} \\\\ & =3025 \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572178187\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate the sum indicated by the notation [latex]\\displaystyle\\sum_{k=1}^{20} (2k+1)[\/latex].<\/p>\n<div id=\"fs-id1170572178190\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8326610\">Hint<\/span><\/p>\n<div id=\"q8326610\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571618983\">Use the rule on sum and powers of integers.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572280444\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572280444\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572280444\">[latex]440[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm19441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19441&theme=oea&iframe_resize_id=ohm19441&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-1116\">\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>5.1 Approximating Areas. <strong>Authored by<\/strong>: Ryan Melton. <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>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":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"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 https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"5.1 Approximating Areas\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1116","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1116\/revisions"}],"predecessor-version":[{"id":1298,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1116\/revisions\/1298"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1113"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1116\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1116"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1116"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1116"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}