{"id":2657,"date":"2024-08-09T22:19:03","date_gmt":"2024-08-09T22:19:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2657"},"modified":"2025-01-27T00:54:38","modified_gmt":"2025-01-27T00:54:38","slug":"module-16-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-16-cheat-sheet\/","title":{"raw":"Sequences and Series: Cheat Sheet","rendered":"Sequences and Series: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<h3><span data-sheets-root=\"1\">Sequences and Their Notations<\/span><\/h3>\r\n<ul>\r\n \t<li>A sequence is a list of numbers, called terms, written in a specific order.<\/li>\r\n \t<li>Explicit formulas define each term of a sequence using the position of the term.<\/li>\r\n \t<li>An explicit formula for the [latex]n\\text{th}[\/latex] term of a sequence can be written by analyzing the pattern of several terms.<\/li>\r\n \t<li>Recursive formulas define each term of a sequence using previous terms.<\/li>\r\n \t<li>Recursive formulas must state the initial term, or terms, of a sequence.<\/li>\r\n \t<li>A set of terms can be written by using a recursive formula.<\/li>\r\n \t<li>A factorial is a mathematical operation that can be defined recursively.<\/li>\r\n \t<li>The factorial of [latex]n[\/latex] is the product of all integers from 1 to [latex]n[\/latex]<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Arithmetic Sequences<\/span><\/h3>\r\n<ul>\r\n \t<li>An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.<\/li>\r\n \t<li>The constant between two consecutive terms is called the common difference.<\/li>\r\n \t<li>The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.<\/li>\r\n \t<li>The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.<\/li>\r\n \t<li>A recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\\ge 2[\/latex].<\/li>\r\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\r\n \t<li>An explicit formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\r\n \t<li>An explicit formula can be used to find the number of terms in a sequence.<\/li>\r\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[\/latex].<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Geometric Sequences<\/span><\/h3>\r\n<ul>\r\n \t<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\r\n \t<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\r\n \t<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\r\n \t<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\r\n \t<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\r\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\r\n \t<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\r\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Series and Their Notations<\/span><\/h3>\r\n<ul>\r\n \t<li>The sum of the terms in a sequence is called a series.<\/li>\r\n \t<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.<\/li>\r\n \t<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\r\n \t<li>The sum of the first [latex]n[\/latex] terms of an arithmetic series can be found using a formula.<\/li>\r\n \t<li>The sum of the terms in a geometric sequence is called a geometric series.<\/li>\r\n \t<li>The sum of the first [latex]n[\/latex] terms of a geometric series can be found using a formula.<\/li>\r\n \t<li>The sum of an infinite series exists if the series is geometric with [latex]-1&lt;r&lt;1[\/latex].<\/li>\r\n \t<li>If the sum of an infinite series exists, it can be found using a formula.<\/li>\r\n \t<li>An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<table style=\"width: 99.5749%;\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>Formula for a factorial<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]\\begin{align}0!&amp;=1\\\\ 1!&amp;=1\\\\ n!&amp;=n\\left(n - 1\\right)\\left(n - 2\\right)\\cdots \\left(2\\right)\\left(1\\right)\\text{, for }n\\ge 2\\end{align}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>recursive formula for nth term of an arithmetic sequence<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{n - 1}+d \\text{ for } n\\ge 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>explicit formula for nth term of an arithmetic sequence<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>sum of the first [latex]n[\/latex]<\/strong>\r\n<strong>terms of an arithmetic series<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>sum of the first [latex]n[\/latex]<\/strong>\r\n<strong>terms of a geometric series<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} , r\\ne 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8782%;\"><strong>sum of an infinite geometric series with [latex]-1&lt;r&lt;1[\/latex]<\/strong><\/td>\r\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{{a}_{1}}{1-r} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165131290028\" class=\"definition\">\r\n \t<dt><strong>annuity<\/strong><\/dt>\r\n \t<dd>an investment in which the purchaser makes a sequence of periodic, equal payments<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290024\" class=\"definition\">\r\n \t<dt><strong>arithmetic sequence<\/strong><\/dt>\r\n \t<dd>a sequence in which the difference between any two consecutive terms is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290029\" class=\"definition\">\r\n \t<dt><strong>arithmetic series<\/strong><\/dt>\r\n \t<dd>the sum of the terms in an arithmetic sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290025\" class=\"definition\">\r\n \t<dt><strong>common difference<\/strong><\/dt>\r\n \t<dd>the difference between any two consecutive terms in an arithmetic sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290026\" class=\"definition\">\r\n \t<dt><strong>common ratio<\/strong><\/dt>\r\n \t<dd>the ratio between any two consecutive terms in a geometric sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290030\" class=\"definition\">\r\n \t<dt><strong>diverge<\/strong><\/dt>\r\n \t<dd>a series is said to diverge if the sum is not a real number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290016\" class=\"definition\">\r\n \t<dt><strong>explicit formula<\/strong><\/dt>\r\n \t<dd>a formula that defines each term of a sequence in terms of its position in the sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290017\" class=\"definition\">\r\n \t<dt><strong>finite sequence<\/strong><\/dt>\r\n \t<dd>a function whose domain consists of a finite subset of the positive integers [latex]\\left\\{1,2,\\dots n\\right\\}[\/latex] for some positive integer [latex]n[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290027\" class=\"definition\">\r\n \t<dt><strong>geometric sequence<\/strong><\/dt>\r\n \t<dd>a sequence in which the ratio of a term to a previous term is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290031\" class=\"definition\">\r\n \t<dt><strong>geometric series<\/strong><\/dt>\r\n \t<dd>the sum of the terms in a geometric sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290018\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165131290032\" class=\"definition\">\r\n \t<dt><strong>index of summation<\/strong><\/dt>\r\n \t<dd>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<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt><strong>infinite sequence<\/strong><\/dt>\r\n \t<dd>a function whose domain is the set of positive integers<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290033\" class=\"definition\">\r\n \t<dt><strong>infinite series<\/strong><\/dt>\r\n \t<dd>the sum of the terms in an infinite sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290034\" class=\"definition\">\r\n \t<dt><strong>lower limit of summation<\/strong><\/dt>\r\n \t<dd>the number used in the explicit formula to find the first term in a series<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290019\" class=\"definition\">\r\n \t<dt><strong>[latex]n[\/latex] factorial<\/strong><\/dt>\r\n \t<dd>the product of all the positive integers from 1 to [latex]n[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290035\" class=\"definition\">\r\n \t<dt><strong>[latex]n[\/latex]th partial sum<\/strong><\/dt>\r\n \t<dd>the sum of the first [latex]n[\/latex] terms of a sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290020\" class=\"definition\">\r\n \t<dt><strong>[latex]n[\/latex]th term of a sequence<\/strong><\/dt>\r\n \t<dd>a formula for the general term of a sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290021\" class=\"definition\">\r\n \t<dt><strong>recursive formula<\/strong><\/dt>\r\n \t<dd>a formula that defines each term of a sequence using previous term(s)<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290022\" class=\"definition\">\r\n \t<dt><strong>sequence<\/strong><\/dt>\r\n \t<dd>a function whose domain is a subset of the positive integers<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290036\" class=\"definition\">\r\n \t<dt><strong>series<\/strong><\/dt>\r\n \t<dd>the sum of the terms in a sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290037\" class=\"definition\">\r\n \t<dt><strong>summation notation<\/strong><\/dt>\r\n \t<dd>a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290023\" class=\"definition\">\r\n \t<dt><strong>term<\/strong><\/dt>\r\n \t<dd>a number in a sequence<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290038\" class=\"definition\">\r\n \t<dt><strong>upper limit of summation<\/strong><\/dt>\r\n \t<dd>the number used in the explicit formula to find the last term in a series<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<h3><span data-sheets-root=\"1\">Sequences and Their Notations<\/span><\/h3>\n<ul>\n<li>A sequence is a list of numbers, called terms, written in a specific order.<\/li>\n<li>Explicit formulas define each term of a sequence using the position of the term.<\/li>\n<li>An explicit formula for the [latex]n\\text{th}[\/latex] term of a sequence can be written by analyzing the pattern of several terms.<\/li>\n<li>Recursive formulas define each term of a sequence using previous terms.<\/li>\n<li>Recursive formulas must state the initial term, or terms, of a sequence.<\/li>\n<li>A set of terms can be written by using a recursive formula.<\/li>\n<li>A factorial is a mathematical operation that can be defined recursively.<\/li>\n<li>The factorial of [latex]n[\/latex] is the product of all integers from 1 to [latex]n[\/latex]<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Arithmetic Sequences<\/span><\/h3>\n<ul>\n<li>An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.<\/li>\n<li>The constant between two consecutive terms is called the common difference.<\/li>\n<li>The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.<\/li>\n<li>The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.<\/li>\n<li>A recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\\ge 2[\/latex].<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n<li>An explicit formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<li>An explicit formula can be used to find the number of terms in a sequence.<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[\/latex].<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Geometric Sequences<\/span><\/h3>\n<ul>\n<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\n<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\n<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Series and Their Notations<\/span><\/h3>\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<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\n<li>The sum of the first [latex]n[\/latex] terms of an arithmetic series can be found using a formula.<\/li>\n<li>The sum of the terms in a geometric sequence is called a geometric series.<\/li>\n<li>The sum of the first [latex]n[\/latex] terms of a geometric series can be found using a formula.<\/li>\n<li>The sum of an infinite series exists if the series is geometric with [latex]-1<r<1[\/latex].<\/li>\n<li>If the sum of an infinite series exists, it can be found using a formula.<\/li>\n<li>An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<table style=\"width: 99.5749%;\" summary=\"..\">\n<tbody>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>Formula for a factorial<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]\\begin{align}0!&=1\\\\ 1!&=1\\\\ n!&=n\\left(n - 1\\right)\\left(n - 2\\right)\\cdots \\left(2\\right)\\left(1\\right)\\text{, for }n\\ge 2\\end{align}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>recursive formula for nth term of an arithmetic sequence<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{n - 1}+d \\text{ for } n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>explicit formula for nth term of an arithmetic sequence<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>sum of the first [latex]n[\/latex]<\/strong><br \/>\n<strong>terms of an arithmetic series<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>sum of the first [latex]n[\/latex]<\/strong><br \/>\n<strong>terms of a geometric series<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} , r\\ne 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8782%;\"><strong>sum of an infinite geometric series with [latex]-1<r<1[\/latex]<\/strong><\/td>\n<td style=\"width: 93.5599%;\">[latex]{S}_{n}=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165131290028\" class=\"definition\">\n<dt><strong>annuity<\/strong><\/dt>\n<dd>an investment in which the purchaser makes a sequence of periodic, equal payments<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290024\" class=\"definition\">\n<dt><strong>arithmetic sequence<\/strong><\/dt>\n<dd>a sequence in which the difference between any two consecutive terms is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290029\" class=\"definition\">\n<dt><strong>arithmetic series<\/strong><\/dt>\n<dd>the sum of the terms in an arithmetic sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290025\" class=\"definition\">\n<dt><strong>common difference<\/strong><\/dt>\n<dd>the difference between any two consecutive terms in an arithmetic sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290026\" class=\"definition\">\n<dt><strong>common ratio<\/strong><\/dt>\n<dd>the ratio between any two consecutive terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290030\" class=\"definition\">\n<dt><strong>diverge<\/strong><\/dt>\n<dd>a series is said to diverge if the sum is not a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290016\" class=\"definition\">\n<dt><strong>explicit formula<\/strong><\/dt>\n<dd>a formula that defines each term of a sequence in terms of its position in the sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290017\" class=\"definition\">\n<dt><strong>finite sequence<\/strong><\/dt>\n<dd>a function whose domain consists of a finite subset of the positive integers [latex]\\left\\{1,2,\\dots n\\right\\}[\/latex] for some positive integer [latex]n[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290027\" class=\"definition\">\n<dt><strong>geometric sequence<\/strong><\/dt>\n<dd>a sequence in which the ratio of a term to a previous term is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290031\" class=\"definition\">\n<dt><strong>geometric series<\/strong><\/dt>\n<dd>the sum of the terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290018\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>index of summation<\/strong><\/dt>\n<dd>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<\/dd>\n<\/dl>\n<p> \t<strong>infinite sequence<\/strong><br \/>\n \ta function whose domain is the set of positive integers<\/p>\n<dl id=\"fs-id1165131290033\" class=\"definition\">\n<dt><strong>infinite series<\/strong><\/dt>\n<dd>the sum of the terms in an infinite sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290034\" class=\"definition\">\n<dt><strong>lower limit of summation<\/strong><\/dt>\n<dd>the number used in the explicit formula to find the first term in a series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290019\" class=\"definition\">\n<dt><strong>[latex]n[\/latex] factorial<\/strong><\/dt>\n<dd>the product of all the positive integers from 1 to [latex]n[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290035\" class=\"definition\">\n<dt><strong>[latex]n[\/latex]th partial sum<\/strong><\/dt>\n<dd>the sum of the first [latex]n[\/latex] terms of a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290020\" class=\"definition\">\n<dt><strong>[latex]n[\/latex]th term of a sequence<\/strong><\/dt>\n<dd>a formula for the general term of a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290021\" class=\"definition\">\n<dt><strong>recursive formula<\/strong><\/dt>\n<dd>a formula that defines each term of a sequence using previous term(s)<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290022\" class=\"definition\">\n<dt><strong>sequence<\/strong><\/dt>\n<dd>a function whose domain is a subset of the positive integers<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290036\" class=\"definition\">\n<dt><strong>series<\/strong><\/dt>\n<dd>the sum of the terms in a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290037\" class=\"definition\">\n<dt><strong>summation notation<\/strong><\/dt>\n<dd>a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290023\" class=\"definition\">\n<dt><strong>term<\/strong><\/dt>\n<dd>a number in a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290038\" class=\"definition\">\n<dt><strong>upper limit of summation<\/strong><\/dt>\n<dd>the number used in the explicit formula to find the last term in a 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