{"id":374,"date":"2025-02-13T19:44:25","date_gmt":"2025-02-13T19:44:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-background-youll-need-2\/"},"modified":"2025-02-13T19:44:25","modified_gmt":"2025-02-13T19:44:25","slug":"introduction-to-integration-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-background-youll-need-2\/","title":{"raw":"Introduction to Integration: Background You'll Need 2","rendered":"Introduction to Integration: Background You&#8217;ll Need 2"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Use sigma notation to add up integers and their powers<\/li>\n<\/ul>\n<\/section>\n<h2>Sigma Notation<\/h2>\n<p id=\"fs-id1170572444485\">To simplify writing lengthy sums, we use sigma notation (summation notation). The Greek letter [latex]\u03a3[\/latex] represents the sum of values. For example, if we want to add all the integers from [latex]1[\/latex] to [latex]20[\/latex], instead of writing out<\/p>\n<center>[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[\/latex]<\/center>\n<p>we can use sigma notation:<\/p>\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex].<\/div>\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 id=\"fs-id1170572108202\">where [latex]a_i[\/latex] are the terms being added and [latex]i[\/latex] is the <span class=\"no-emphasis\"><em>index<\/em><\/span>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>sigma notation<\/h3>\n<p>Sigma notation uses the Greek letter sigma ([latex]\u2211[\/latex]) to represent the sum of a series of terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/p>\n<p>Each term [latex]a_i[\/latex] is evaluated for all integer values of [latex]i[\/latex] from the lower limit to the upper limit, and then all these values are added together.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Evaluate Sigma Notation<\/strong><\/p>\n<ol>\n\t<li><strong>Identify the Index and Limits: <\/strong>Locate the index variable [latex]i[\/latex], the starting value (often [latex]1[\/latex]), and the upper limit [latex]n[\/latex].<\/li>\n\t<li><strong>Determine the Term Expression: <\/strong>Identify the term [latex]a_i[\/latex] that you will be summing.<\/li>\n\t<li><strong>Evaluate Each Term:&nbsp;<\/strong>Substitute each integer value from the starting value to the upper limit into the term expression.<\/li>\n\t<li><strong>Sum the Evaluated Terms:&nbsp;<\/strong>Add up all the evaluated terms to get the final sum.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>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].<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>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.<\/p>\n<p>Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\n<\/section>\n<p id=\"fs-id1170572472220\">Let\u2019s try a couple of examples of using sigma notation.<\/p>\n<section class=\"textbox example\">\n<ol id=\"fs-id1170571671542\" style=\"list-style-type: lower-alpha;\">\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>\n\t<li>Write the sum in sigma notation:<br>\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\n[reveal-answer q=\"fs-id1170572346829\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572346829\"]\n\n<ol id=\"fs-id1170572346829\" style=\"list-style-type: lower-alpha;\">\n\t<li>Write<br>\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>\n<\/li>\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>\n<\/ol>\n<p>Watch the following video to see the worked solution to Example: Using Sigma Notation.<\/p>\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\" data-mce-fragment=\"1\"><\/iframe><\/center>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.\n\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 \"5.1 Approximating Areas\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<p id=\"fs-id1170572108042\">The following rules summarize key properties of sigma notation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">Properties of Sigma Notation<\/h3>\n<div>\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.<\/p>\n<p>&nbsp;<\/p>\n<p>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\t<li>\n<div id=\"fs-id1170572294298\" class=\"equation\">[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<\/div>\n<\/li>\n\t<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\t<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\t<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\t<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<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr>\n<p id=\"fs-id1170572453570\">Let's prove properties 2 and 3, and leave the proof of the other properties for the examples.<\/p>\n<p id=\"fs-id1167794031899\"><strong>Proof of Property 2:&nbsp;<\/strong><\/p>\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>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793969731\"><strong>Proof of Property 3:&nbsp;<\/strong><\/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) &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>\n<p id=\"fs-id1170572110150\">[latex]_\\blacksquare[\/latex]<\/p>\n<\/section>\n<p id=\"fs-id1170572451163\">Here are a few more formulas that simplify the summation process for frequently encountered functions. These rules, which apply to sums and powers of integers, will be used in the upcoming examples.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">Sums and Powers of Integers<\/h3>\n<ol id=\"fs-id1170572608710\">\n\t<li>The sum of [latex]n[\/latex] integers is given by<br>\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\t<li>The sum of consecutive integers squared is given by<br>\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\t<li>The sum of consecutive integers cubed is given by<br>\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<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572297312\">Write using sigma notation and evaluate:<\/p>\n<ol id=\"fs-id1170571635786\" style=\"list-style-type: lower-alpha;\">\n\t<li>The sum of the terms [latex](i-3)^2[\/latex] for [latex]i=1,2,\\cdots,200[\/latex].<\/li>\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>\n<\/ol>\n\n[reveal-answer q=\"fs-id1170572088097\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572088097\"]\n\n<ol id=\"fs-id1170572088097\" style=\"list-style-type: lower-alpha;\">\n\t<li>Multiplying out [latex](i-3)^2[\/latex], we can break the expression into three terms.<br>\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>\n<\/li>\n\t<li>Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.<br>\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>\n<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution this example.<\/p>\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\" data-mce-fragment=\"1\"><\/iframe><\/center>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.\n\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 \"5.1 Approximating Areas\" here (opens in new window)<\/a>.\/p&gt;<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the sum of the values of [latex]4+3i[\/latex] for [latex]i=1,2,\\cdots,100[\/latex].<\/p>\n<p>[reveal-answer q=\"fs-id1170572292992\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572292992\"]<\/p>\n<p>[latex]15,550[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the sum indicated by the notation [latex]\\displaystyle\\sum_{k=1}^{20} (2k+1)[\/latex].<\/p>\n<p>[reveal-answer q=\"fs-id1170572280444\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572280444\"]<\/p>\n<p id=\"fs-id1170572280444\">[latex]440[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]288430[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use sigma notation to add up integers and their powers<\/li>\n<\/ul>\n<\/section>\n<h2>Sigma Notation<\/h2>\n<p id=\"fs-id1170572444485\">To simplify writing lengthy sums, we use sigma notation (summation notation). The Greek letter [latex]\u03a3[\/latex] represents the sum of values. For example, if we want to add all the integers from [latex]1[\/latex] to [latex]20[\/latex], instead of writing out<\/p>\n<div 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>we can use sigma notation:<\/p>\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex].<\/div>\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 id=\"fs-id1170572108202\">where [latex]a_i[\/latex] are the terms being added and [latex]i[\/latex] is the <span class=\"no-emphasis\"><em>index<\/em><\/span>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>sigma notation<\/h3>\n<p>Sigma notation uses the Greek letter sigma ([latex]\u2211[\/latex]) to represent the sum of a series of terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/p>\n<p>Each term [latex]a_i[\/latex] is evaluated for all integer values of [latex]i[\/latex] from the lower limit to the upper limit, and then all these values are added together.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Evaluate Sigma Notation<\/strong><\/p>\n<ol>\n<li><strong>Identify the Index and Limits: <\/strong>Locate the index variable [latex]i[\/latex], the starting value (often [latex]1[\/latex]), and the upper limit [latex]n[\/latex].<\/li>\n<li><strong>Determine the Term Expression: <\/strong>Identify the term [latex]a_i[\/latex] that you will be summing.<\/li>\n<li><strong>Evaluate Each Term:&nbsp;<\/strong>Substitute each integer value from the starting value to the upper limit into the term expression.<\/li>\n<li><strong>Sum the Evaluated Terms:&nbsp;<\/strong>Add up all the evaluated terms to get the final sum.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>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].<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>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.<\/p>\n<p>Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\n<\/section>\n<p id=\"fs-id1170572472220\">Let\u2019s try a couple of examples of using sigma notation.<\/p>\n<section class=\"textbox example\">\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 class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572346829\">Show Solution<\/button><\/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<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\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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<\/section>\n<p id=\"fs-id1170572108042\">The following rules summarize key properties of sigma notation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">Properties of Sigma Notation<\/h3>\n<div>\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.<\/p>\n<p>&nbsp;<\/p>\n<p>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<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr \/>\n<p id=\"fs-id1170572453570\">Let&#8217;s prove properties 2 and 3, and leave the proof of the other properties for the examples.<\/p>\n<p id=\"fs-id1167794031899\"><strong>Proof of Property 2:&nbsp;<\/strong><\/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\"><strong>Proof of Property 3:&nbsp;<\/strong><\/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<\/section>\n<p id=\"fs-id1170572451163\">Here are a few more formulas that simplify the summation process for frequently encountered functions. These rules, which apply to sums and powers of integers, will be used in the upcoming examples.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">Sums and Powers of Integers<\/h3>\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<\/section>\n<section class=\"textbox example\">\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 class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572088097\">Show Solution<\/button><\/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<p>Watch the following video to see the worked solution this example.<\/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\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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>.\/p&gt;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the sum of the values of [latex]4+3i[\/latex] for [latex]i=1,2,\\cdots,100[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572292992\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572292992\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]15,550[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the sum indicated by the notation [latex]\\displaystyle\\sum_{k=1}^{20} (2k+1)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572280444\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288430\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288430&theme=lumen&iframe_resize_id=ohm288430&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":371,"module-header":"","content_attributions":[],"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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/374"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/374\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/374\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=374"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=374"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=374"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}