{"id":2285,"date":"2021-09-21T15:11:49","date_gmt":"2021-09-21T15:11:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-comparison-tests\/"},"modified":"2022-04-19T20:51:31","modified_gmt":"2022-04-19T20:51:31","slug":"skills-review-for-comparison-tests","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-comparison-tests\/","title":{"raw":"Skills Review for Comparison Tests","rendered":"Skills Review for Comparison Tests"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplifying expressions using the Quotient Property of Exponents<\/li>\r\n \t<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\r\n \t<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Comparison Tests section, we will explore some more methods that can be used to determine whether an infinite series diverges or converges. Here we will review the quotient rule for exponents, how to take limits at infinity, and L'Hopital's Rule.\r\n<h2>The Quotient Rule for Exponents<\/h2>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m&gt;n[\/latex], the quotient rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Quotient Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"717838\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"717838\"]\r\n\r\nUse the quotient rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=mom60[\/embed]\r\n\r\n<\/div>\r\n<h2>Take Limits at Infinity<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>\r\n<h2>Infinite Limits at Infinity<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>\r\n<h2>Apply L'H\u00f4pital's Rule<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplifying expressions using the Quotient Property of Exponents<\/li>\n<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\n<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule<\/li>\n<\/ul>\n<\/div>\n<p>In the Comparison Tests section, we will explore some more methods that can be used to determine whether an infinite series diverges or converges. Here we will review the quotient rule for exponents, how to take limits at infinity, and L&#8217;Hopital&#8217;s Rule.<\/p>\n<h2>The Quotient Rule for Exponents<\/h2>\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m>n[\/latex], the quotient rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717838\">Show Solution<\/span><\/p>\n<div id=\"q717838\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the quotient rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109745\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&#38;theme=oea&#38;iframe_resize_id=ohm109745&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Take Limits at Infinity<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n<h2>Infinite Limits at Infinity<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n<h2>Apply L&#8217;H\u00f4pital&#8217;s Rule<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2285\">\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>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen 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