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<li><a href="./">M112: Differential Calculus</a></li>

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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Welcome</a></li>
<li class="chapter" data-level="1" data-path="functions-and-models.html"><a href="functions-and-models.html"><i class="fa fa-check"></i><b>1</b> Functions and Models</a>
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<li class="chapter" data-level="1.1" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html"><i class="fa fa-check"></i><b>1.1</b> Four Ways to Represent a Function</a>
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<li class="chapter" data-level="1.1.5" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#putting-it-all-together"><i class="fa fa-check"></i><b>1.1.5</b> Putting it All Together</a></li>
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<li class="chapter" data-level="1.2.2" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#polynomial-functions"><i class="fa fa-check"></i><b>1.2.2</b> Polynomial Functions</a></li>
<li class="chapter" data-level="1.2.3" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#power-functions"><i class="fa fa-check"></i><b>1.2.3</b> Power Functions</a></li>
<li class="chapter" data-level="1.2.4" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#rational-functions"><i class="fa fa-check"></i><b>1.2.4</b> Rational Functions</a></li>
<li class="chapter" data-level="1.2.5" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#algebraic-functions"><i class="fa fa-check"></i><b>1.2.5</b> Algebraic Functions</a></li>
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<li class="chapter" data-level="1.2.7" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#exponential-functions"><i class="fa fa-check"></i><b>1.2.7</b> Exponential Functions</a></li>
<li class="chapter" data-level="1.2.8" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#logarithmic-functions"><i class="fa fa-check"></i><b>1.2.8</b> Logarithmic Functions</a></li>
<li class="chapter" data-level="1.2.9" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#putting-it-all-together-1"><i class="fa fa-check"></i><b>1.2.9</b> Putting it All Together</a></li>
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<li class="chapter" data-level="1.3" data-path="new-functions-from-old-functions.html"><a href="new-functions-from-old-functions.html"><i class="fa fa-check"></i><b>1.3</b> New Functions from Old Functions</a>
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<li class="chapter" data-level="1.4.2" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#applications-of-exponential-functions"><i class="fa fa-check"></i><b>1.4.2</b> Applications of Exponential Functions</a></li>
<li class="chapter" data-level="1.4.3" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#the-number-e-and-the-natural-exponential-function"><i class="fa fa-check"></i><b>1.4.3</b> The Number <span class="math inline">\(e\)</span> and the Natural Exponential Function</a></li>
<li class="chapter" data-level="1.4.4" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#graph-transformations-of-exponentials"><i class="fa fa-check"></i><b>1.4.4</b> Graph Transformations of Exponentials</a></li>
<li class="chapter" data-level="1.4.5" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#putting-it-all-together-3"><i class="fa fa-check"></i><b>1.4.5</b> Putting It All Together</a></li>
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<li class="chapter" data-level="1.5" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html"><i class="fa fa-check"></i><b>1.5</b> Inverse Functions and Logarithms</a>
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<li class="chapter" data-level="1.5.1" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#graphs-of-inverse-functions"><i class="fa fa-check"></i><b>1.5.1</b> Graphs of Inverse Functions</a></li>
<li class="chapter" data-level="1.5.2" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#logarithmic-functions-as-inverses"><i class="fa fa-check"></i><b>1.5.2</b> Logarithmic Functions as Inverses</a></li>
<li class="chapter" data-level="1.5.3" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#solving-exponential-and-log-equations"><i class="fa fa-check"></i><b>1.5.3</b> Solving Exponential and Log Equations</a></li>
<li class="chapter" data-level="1.5.4" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#inverse-trigonometric-functions"><i class="fa fa-check"></i><b>1.5.4</b> Inverse Trigonometric Functions</a></li>
<li class="chapter" data-level="1.5.5" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#pulling-it-all-together"><i class="fa fa-check"></i><b>1.5.5</b> Pulling It All Together</a></li>
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<li class="chapter" data-level="2" data-path="limits-and-derivatives.html"><a href="limits-and-derivatives.html"><i class="fa fa-check"></i><b>2</b> Limits and Derivatives</a>
<ul>
<li class="chapter" data-level="2.1" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html"><i class="fa fa-check"></i><b>2.1</b> The Tangent and Velocity Problems</a>
<ul>
<li class="chapter" data-level="2.1.1" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#the-tangent-problem"><i class="fa fa-check"></i><b>2.1.1</b> The Tangent Problem</a></li>
<li class="chapter" data-level="2.1.2" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#tangents-from-experimental-data"><i class="fa fa-check"></i><b>2.1.2</b> Tangents from Experimental Data</a></li>
<li class="chapter" data-level="2.1.3" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#the-velocity-problem"><i class="fa fa-check"></i><b>2.1.3</b> The Velocity Problem</a></li>
<li class="chapter" data-level="2.1.4" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#connection-between-tangents-and-velocity"><i class="fa fa-check"></i><b>2.1.4</b> Connection Between Tangents and Velocity</a></li>
<li class="chapter" data-level="2.1.5" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#putting-it-all-together-4"><i class="fa fa-check"></i><b>2.1.5</b> Putting It All Together</a></li>
<li class="chapter" data-level="2.1.6" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#conceptual-takeaways-5"><i class="fa fa-check"></i><b>2.1.6</b> Conceptual Takeaways</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html"><i class="fa fa-check"></i><b>2.2</b> The Limit of a Function</a>
<ul>
<li class="chapter" data-level="2.2.1" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#intuitive-idea-of-a-limit"><i class="fa fa-check"></i><b>2.2.1</b> Intuitive Idea of a Limit</a></li>
<li class="chapter" data-level="2.2.2" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#numerical-approach-to-limits"><i class="fa fa-check"></i><b>2.2.2</b> Numerical Approach to Limits</a></li>
<li class="chapter" data-level="2.2.3" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#one-sided-limits"><i class="fa fa-check"></i><b>2.2.3</b> One-Sided Limits</a></li>
<li class="chapter" data-level="2.2.4" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#infinite-limits"><i class="fa fa-check"></i><b>2.2.4</b> Infinite Limits</a></li>
<li class="chapter" data-level="2.2.5" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#vertical-asymptotes"><i class="fa fa-check"></i><b>2.2.5</b> Vertical Asymptotes</a></li>
<li class="chapter" data-level="2.2.6" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#putting-it-all-together-5"><i class="fa fa-check"></i><b>2.2.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html"><i class="fa fa-check"></i><b>2.3</b> Calculating Limits Using the Limit Laws</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#limit-laws"><i class="fa fa-check"></i><b>2.3.1</b> Limit Laws</a></li>
<li class="chapter" data-level="2.3.2" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#the-squeeze-theorem-sandwich-theorem"><i class="fa fa-check"></i><b>2.3.2</b> The Squeeze Theorem (Sandwich Theorem)</a></li>
<li class="chapter" data-level="2.3.3" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#putting-it-all-together-6"><i class="fa fa-check"></i><b>2.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="the-precise-definition-of-a-limit.html"><a href="the-precise-definition-of-a-limit.html"><i class="fa fa-check"></i><b>2.4</b> The Precise Definition of a Limit</a></li>
<li class="chapter" data-level="2.5" data-path="continuity.html"><a href="continuity.html"><i class="fa fa-check"></i><b>2.5</b> Continuity</a>
<ul>
<li class="chapter" data-level="2.5.1" data-path="continuity.html"><a href="continuity.html#types-of-discontinuities"><i class="fa fa-check"></i><b>2.5.1</b> Types of Discontinuities</a></li>
<li class="chapter" data-level="2.5.2" data-path="continuity.html"><a href="continuity.html#one-sided-continuity"><i class="fa fa-check"></i><b>2.5.2</b> One-Sided Continuity</a></li>
<li class="chapter" data-level="2.5.3" data-path="continuity.html"><a href="continuity.html#composite-functions"><i class="fa fa-check"></i><b>2.5.3</b> Composite Functions</a></li>
<li class="chapter" data-level="2.5.4" data-path="continuity.html"><a href="continuity.html#intermediate-value-theorem-ivt"><i class="fa fa-check"></i><b>2.5.4</b> Intermediate Value Theorem (IVT)</a></li>
<li class="chapter" data-level="2.5.5" data-path="continuity.html"><a href="continuity.html#putting-it-all-together-7"><i class="fa fa-check"></i><b>2.5.5</b> Putting It All Together</a></li>
<li class="chapter" data-level="2.5.6" data-path="continuity.html"><a href="continuity.html#skills-you-should-be-able-to-do-7"><i class="fa fa-check"></i><b>2.5.6</b> Skills You Should Be Able To Do</a></li>
<li class="chapter" data-level="2.5.7" data-path="continuity.html"><a href="continuity.html#problems-6"><i class="fa fa-check"></i><b>2.5.7</b> Problems</a></li>
</ul></li>
<li class="chapter" data-level="2.6" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html"><i class="fa fa-check"></i><b>2.6</b> Limits at Infinity; Horizontal Asymptotes</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#limits-at-infinity"><i class="fa fa-check"></i><b>2.6.1</b> Limits at Infinity</a></li>
<li class="chapter" data-level="2.6.2" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#infinite-limits-at-infinity"><i class="fa fa-check"></i><b>2.6.2</b> Infinite Limits at Infinity</a></li>
<li class="chapter" data-level="2.6.3" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#graph-interpretation"><i class="fa fa-check"></i><b>2.6.3</b> Graph Interpretation</a></li>
<li class="chapter" data-level="2.6.4" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#putting-it-all-together-8"><i class="fa fa-check"></i><b>2.6.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.7" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html"><i class="fa fa-check"></i><b>2.7</b> Derivatives and Rates of Change</a>
<ul>
<li class="chapter" data-level="2.7.1" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#tangent-lines-and-limits"><i class="fa fa-check"></i><b>2.7.1</b> Tangent Lines and Limits</a></li>
<li class="chapter" data-level="2.7.2" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#derivatives"><i class="fa fa-check"></i><b>2.7.2</b> Derivatives</a></li>
<li class="chapter" data-level="2.7.3" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#tangent-line-using-derivatives"><i class="fa fa-check"></i><b>2.7.3</b> Tangent Line Using Derivatives</a></li>
<li class="chapter" data-level="2.7.4" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#rates-of-change"><i class="fa fa-check"></i><b>2.7.4</b> Rates of Change</a></li>
<li class="chapter" data-level="2.7.5" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#putting-it-all-together-9"><i class="fa fa-check"></i><b>2.7.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.8" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html"><i class="fa fa-check"></i><b>2.8</b> The Derivative as a Function</a>
<ul>
<li class="chapter" data-level="2.8.1" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#continuity-vs-differentiability"><i class="fa fa-check"></i><b>2.8.1</b> Continuity vs Differentiability</a></li>
<li class="chapter" data-level="2.8.2" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#higher-derivatives"><i class="fa fa-check"></i><b>2.8.2</b> Higher Derivatives</a></li>
<li class="chapter" data-level="2.8.3" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#putting-it-all-together-10"><i class="fa fa-check"></i><b>2.8.3</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="differentiation-rules.html"><a href="differentiation-rules.html"><i class="fa fa-check"></i><b>3</b> Differentiation Rules</a>
<ul>
<li class="chapter" data-level="3.1" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html"><i class="fa fa-check"></i><b>3.1</b> Derivatives of Polynomials and Exponential Functions</a>
<ul>
<li class="chapter" data-level="3.1.1" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#new-derivatives-from-old"><i class="fa fa-check"></i><b>3.1.1</b> New Derivatives from Old</a></li>
<li class="chapter" data-level="3.1.2" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#exponential-functions-2"><i class="fa fa-check"></i><b>3.1.2</b> Exponential Functions</a></li>
<li class="chapter" data-level="3.1.3" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#putting-it-all-together-11"><i class="fa fa-check"></i><b>3.1.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.2" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html"><i class="fa fa-check"></i><b>3.2</b> The Product and Quotient Rules</a>
<ul>
<li class="chapter" data-level="3.2.1" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#the-product-rule"><i class="fa fa-check"></i><b>3.2.1</b> The Product Rule</a></li>
<li class="chapter" data-level="3.2.2" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#the-quotient-rule"><i class="fa fa-check"></i><b>3.2.2</b> The Quotient Rule</a></li>
<li class="chapter" data-level="3.2.3" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#putting-it-all-together-12"><i class="fa fa-check"></i><b>3.2.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.3" data-path="derivatives-of-trigonometric-functions.html"><a href="derivatives-of-trigonometric-functions.html"><i class="fa fa-check"></i><b>3.3</b> Derivatives of Trigonometric Functions</a>
<ul>
<li class="chapter" data-level="3.3.1" data-path="derivatives-of-trigonometric-functions.html"><a href="derivatives-of-trigonometric-functions.html#putting-it-all-together-13"><i class="fa fa-check"></i><b>3.3.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.4" data-path="the-chain-rule.html"><a href="the-chain-rule.html"><i class="fa fa-check"></i><b>3.4</b> The Chain Rule</a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="the-chain-rule.html"><a href="the-chain-rule.html#putting-it-all-together-14"><i class="fa fa-check"></i><b>3.4.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html"><i class="fa fa-check"></i><b>3.5</b> Implicit Differentiation</a>
<ul>
<li class="chapter" data-level="3.5.1" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#horizontal-tangent-condition"><i class="fa fa-check"></i><b>3.5.1</b> Horizontal Tangent Condition</a></li>
<li class="chapter" data-level="3.5.2" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#inverse-trigonometric-functions-via-implicit-differentiation"><i class="fa fa-check"></i><b>3.5.2</b> Inverse Trigonometric Functions via Implicit Differentiation</a></li>
<li class="chapter" data-level="3.5.3" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#putting-it-all-together-15"><i class="fa fa-check"></i><b>3.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.6" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html"><i class="fa fa-check"></i><b>3.6</b> Derivatives of Logarithmic Functions</a>
<ul>
<li class="chapter" data-level="3.6.1" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html#logarithmic-differentiation"><i class="fa fa-check"></i><b>3.6.1</b> Logarithmic Differentiation</a></li>
<li class="chapter" data-level="3.6.2" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html#putting-it-all-together-16"><i class="fa fa-check"></i><b>3.6.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.7" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html"><i class="fa fa-check"></i><b>3.7</b> Rates of Change in the Natural and Social Sciences</a>
<ul>
<li class="chapter" data-level="3.7.1" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#physics-applications"><i class="fa fa-check"></i><b>3.7.1</b> Physics Applications</a></li>
<li class="chapter" data-level="3.7.2" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#physics-density-and-current"><i class="fa fa-check"></i><b>3.7.2</b> Physics: Density and Current</a></li>
<li class="chapter" data-level="3.7.3" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#electric-current"><i class="fa fa-check"></i><b>3.7.3</b> Electric Current</a></li>
<li class="chapter" data-level="3.7.4" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#chemistry-applications"><i class="fa fa-check"></i><b>3.7.4</b> Chemistry Applications</a></li>
<li class="chapter" data-level="3.7.5" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#biology-applications"><i class="fa fa-check"></i><b>3.7.5</b> Biology Applications</a></li>
<li class="chapter" data-level="3.7.6" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#economics-applications"><i class="fa fa-check"></i><b>3.7.6</b> Economics Applications</a></li>
<li class="chapter" data-level="3.7.7" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#putting-it-all-together-17"><i class="fa fa-check"></i><b>3.7.7</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.8" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html"><i class="fa fa-check"></i><b>3.8</b> Exponential Growth and Decay</a>
<ul>
<li class="chapter" data-level="3.8.1" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#the-law-of-natural-growth-and-decay"><i class="fa fa-check"></i><b>3.8.1</b> The Law of Natural Growth and Decay</a></li>
<li class="chapter" data-level="3.8.2" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#population-growth-model"><i class="fa fa-check"></i><b>3.8.2</b> Population Growth Model</a></li>
<li class="chapter" data-level="3.8.3" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#newtons-law-of-cooling"><i class="fa fa-check"></i><b>3.8.3</b> Newton’s Law of Cooling</a></li>
<li class="chapter" data-level="3.8.4" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#continuously-compounded-interest"><i class="fa fa-check"></i><b>3.8.4</b> Continuously Compounded Interest</a></li>
<li class="chapter" data-level="3.8.5" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#putting-it-all-together-18"><i class="fa fa-check"></i><b>3.8.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.9" data-path="related-rates.html"><a href="related-rates.html"><i class="fa fa-check"></i><b>3.9</b> Related Rates</a>
<ul>
<li class="chapter" data-level="3.9.1" data-path="related-rates.html"><a href="related-rates.html#putting-it-all-together-19"><i class="fa fa-check"></i><b>3.9.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.10" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html"><i class="fa fa-check"></i><b>3.10</b> Linear Approximations and Differentials</a>
<ul>
<li class="chapter" data-level="3.10.1" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#tangent-line-approximation"><i class="fa fa-check"></i><b>3.10.1</b> Tangent Line Approximation</a></li>
<li class="chapter" data-level="3.10.2" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#differentials"><i class="fa fa-check"></i><b>3.10.2</b> Differentials</a></li>
<li class="chapter" data-level="3.10.3" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#putting-it-all-together-20"><i class="fa fa-check"></i><b>3.10.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.11" data-path="hyperbolic-functions.html"><a href="hyperbolic-functions.html"><i class="fa fa-check"></i><b>3.11</b> Hyperbolic Functions</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="applications-of-differentiation.html"><a href="applications-of-differentiation.html"><i class="fa fa-check"></i><b>4</b> Applications of Differentiation</a>
<ul>
<li class="chapter" data-level="4.1" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html"><i class="fa fa-check"></i><b>4.1</b> Maximum and Minimum Values</a>
<ul>
<li class="chapter" data-level="4.1.1" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#types-of-extreme-values"><i class="fa fa-check"></i><b>4.1.1</b> Types of Extreme Values</a></li>
<li class="chapter" data-level="4.1.2" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#extreme-value-theorem-evt"><i class="fa fa-check"></i><b>4.1.2</b> Extreme Value Theorem (EVT)</a></li>
<li class="chapter" data-level="4.1.3" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#fermats-theorem"><i class="fa fa-check"></i><b>4.1.3</b> Fermat’s Theorem</a></li>
<li class="chapter" data-level="4.1.4" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#critical-numbers"><i class="fa fa-check"></i><b>4.1.4</b> Critical Numbers</a></li>
<li class="chapter" data-level="4.1.5" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#closed-interval-method"><i class="fa fa-check"></i><b>4.1.5</b> Closed Interval Method</a></li>
<li class="chapter" data-level="4.1.6" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#putting-it-all-together-21"><i class="fa fa-check"></i><b>4.1.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.2" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html"><i class="fa fa-check"></i><b>4.2</b> The Mean Value Theorem</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#rolles-theorem"><i class="fa fa-check"></i><b>4.2.1</b> Rolle’s Theorem</a></li>
<li class="chapter" data-level="4.2.2" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#the-mean-value-theorem-1"><i class="fa fa-check"></i><b>4.2.2</b> The Mean Value Theorem</a></li>
<li class="chapter" data-level="4.2.3" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#consequences-of-the-mean-value-theorem"><i class="fa fa-check"></i><b>4.2.3</b> Consequences of the Mean Value Theorem</a></li>
<li class="chapter" data-level="4.2.4" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#putting-it-all-together-22"><i class="fa fa-check"></i><b>4.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html"><i class="fa fa-check"></i><b>4.3</b> How Derivatives Affect the Shape of a Graph</a>
<ul>
<li class="chapter" data-level="4.3.1" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#what-does-f-say-about-f"><i class="fa fa-check"></i><b>4.3.1</b> What Does <span class="math inline">\(f&#39;\)</span> Say About <span class="math inline">\(f\)</span>?</a></li>
<li class="chapter" data-level="4.3.2" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#local-extreme-values"><i class="fa fa-check"></i><b>4.3.2</b> Local Extreme Values</a></li>
<li class="chapter" data-level="4.3.3" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#what-does-f-say-about-f-1"><i class="fa fa-check"></i><b>4.3.3</b> What Does <span class="math inline">\(f&#39;&#39;\)</span> Say About <span class="math inline">\(f\)</span>?</a></li>
<li class="chapter" data-level="4.3.4" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#inflection-points"><i class="fa fa-check"></i><b>4.3.4</b> Inflection Points</a></li>
<li class="chapter" data-level="4.3.5" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#graph-shape-analysis-framework"><i class="fa fa-check"></i><b>4.3.5</b> Graph Shape Analysis Framework</a></li>
<li class="chapter" data-level="4.3.6" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#putting-it-all-together-23"><i class="fa fa-check"></i><b>4.3.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html"><i class="fa fa-check"></i><b>4.4</b> Indeterminate Forms and l’Hospital’s Rule</a>
<ul>
<li class="chapter" data-level="4.4.1" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-forms"><i class="fa fa-check"></i><b>4.4.1</b> Indeterminate Forms</a></li>
<li class="chapter" data-level="4.4.2" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#interpretation"><i class="fa fa-check"></i><b>4.4.2</b> Interpretation</a></li>
<li class="chapter" data-level="4.4.3" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-products-0-cdot-infty"><i class="fa fa-check"></i><b>4.4.3</b> Indeterminate Products <span class="math inline">\(0 \cdot \infty\)</span></a></li>
<li class="chapter" data-level="4.4.4" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-differences-infty---infty"><i class="fa fa-check"></i><b>4.4.4</b> Indeterminate Differences <span class="math inline">\(\infty - \infty\)</span></a></li>
<li class="chapter" data-level="4.4.5" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-powers"><i class="fa fa-check"></i><b>4.4.5</b> Indeterminate Powers</a></li>
<li class="chapter" data-level="4.4.6" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#putting-it-all-together-24"><i class="fa fa-check"></i><b>4.4.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.5" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html"><i class="fa fa-check"></i><b>4.5</b> Summary of Curve Sketching</a>
<ul>
<li class="chapter" data-level="4.5.1" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html#a-structured-framework-for-sketching-curves"><i class="fa fa-check"></i><b>4.5.1</b> A Structured Framework for Sketching Curves</a></li>
<li class="chapter" data-level="4.5.2" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html#putting-it-all-together-25"><i class="fa fa-check"></i><b>4.5.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="graphing-with-calculus-and-calculators.html"><a href="graphing-with-calculus-and-calculators.html"><i class="fa fa-check"></i><b>4.6</b> Graphing with Calculus and Calculators</a></li>
<li class="chapter" data-level="4.7" data-path="optimization-problems.html"><a href="optimization-problems.html"><i class="fa fa-check"></i><b>4.7</b> Optimization Problems</a>
<ul>
<li class="chapter" data-level="4.7.1" data-path="optimization-problems.html"><a href="optimization-problems.html#general-strategy-for-solving-optimization-problems"><i class="fa fa-check"></i><b>4.7.1</b> General Strategy for Solving Optimization Problems</a></li>
<li class="chapter" data-level="4.7.2" data-path="optimization-problems.html"><a href="optimization-problems.html#applications-in-business-and-economics"><i class="fa fa-check"></i><b>4.7.2</b> Applications in Business and Economics</a></li>
<li class="chapter" data-level="4.7.3" data-path="optimization-problems.html"><a href="optimization-problems.html#putting-it-all-together-26"><i class="fa fa-check"></i><b>4.7.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.8" data-path="newtons-method.html"><a href="newtons-method.html"><i class="fa fa-check"></i><b>4.8</b> Newton’s Method</a>
<ul>
<li class="chapter" data-level="4.8.1" data-path="newtons-method.html"><a href="newtons-method.html#derivation-of-the-formula"><i class="fa fa-check"></i><b>4.8.1</b> Derivation of the Formula</a></li>
<li class="chapter" data-level="4.8.2" data-path="newtons-method.html"><a href="newtons-method.html#convergence-and-failure"><i class="fa fa-check"></i><b>4.8.2</b> Convergence and Failure</a></li>
<li class="chapter" data-level="4.8.3" data-path="newtons-method.html"><a href="newtons-method.html#stopping-criterion"><i class="fa fa-check"></i><b>4.8.3</b> Stopping Criterion</a></li>
<li class="chapter" data-level="4.8.4" data-path="newtons-method.html"><a href="newtons-method.html#putting-it-all-together-27"><i class="fa fa-check"></i><b>4.8.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.9" data-path="antiderivatives.html"><a href="antiderivatives.html"><i class="fa fa-check"></i><b>4.9</b> Antiderivatives</a>
<ul>
<li class="chapter" data-level="4.9.1" data-path="antiderivatives.html"><a href="antiderivatives.html#general-antiderivative-theorem"><i class="fa fa-check"></i><b>4.9.1</b> General Antiderivative Theorem</a></li>
<li class="chapter" data-level="4.9.2" data-path="antiderivatives.html"><a href="antiderivatives.html#basic-antidifferentiation-rules"><i class="fa fa-check"></i><b>4.9.2</b> Basic Antidifferentiation Rules</a></li>
<li class="chapter" data-level="4.9.3" data-path="antiderivatives.html"><a href="antiderivatives.html#graphical-interpretation"><i class="fa fa-check"></i><b>4.9.3</b> Graphical Interpretation</a></li>
<li class="chapter" data-level="4.9.4" data-path="antiderivatives.html"><a href="antiderivatives.html#putting-it-all-together-28"><i class="fa fa-check"></i><b>4.9.4</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="5" data-path="integrals.html"><a href="integrals.html"><i class="fa fa-check"></i><b>5</b> Integrals</a>
<ul>
<li class="chapter" data-level="5.1" data-path="areas-and-distances.html"><a href="areas-and-distances.html"><i class="fa fa-check"></i><b>5.1</b> Areas and Distances</a>
<ul>
<li class="chapter" data-level="5.1.1" data-path="areas-and-distances.html"><a href="areas-and-distances.html#the-area-problem"><i class="fa fa-check"></i><b>5.1.1</b> The Area Problem</a></li>
<li class="chapter" data-level="5.1.2" data-path="areas-and-distances.html"><a href="areas-and-distances.html#types-of-riemann-sums"><i class="fa fa-check"></i><b>5.1.2</b> Types of Riemann Sums</a></li>
<li class="chapter" data-level="5.1.3" data-path="areas-and-distances.html"><a href="areas-and-distances.html#the-distance-problem"><i class="fa fa-check"></i><b>5.1.3</b> The Distance Problem</a></li>
<li class="chapter" data-level="5.1.4" data-path="areas-and-distances.html"><a href="areas-and-distances.html#putting-it-all-together-29"><i class="fa fa-check"></i><b>5.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="the-definite-integral.html"><a href="the-definite-integral.html"><i class="fa fa-check"></i><b>5.2</b> The Definite Integral</a>
<ul>
<li class="chapter" data-level="5.2.1" data-path="the-definite-integral.html"><a href="the-definite-integral.html#riemann-sums"><i class="fa fa-check"></i><b>5.2.1</b> Riemann Sums</a></li>
<li class="chapter" data-level="5.2.2" data-path="the-definite-integral.html"><a href="the-definite-integral.html#the-midpoint-rule"><i class="fa fa-check"></i><b>5.2.2</b> The Midpoint Rule</a></li>
<li class="chapter" data-level="5.2.3" data-path="the-definite-integral.html"><a href="the-definite-integral.html#properties-of-the-definite-integral"><i class="fa fa-check"></i><b>5.2.3</b> Properties of the Definite Integral</a></li>
<li class="chapter" data-level="5.2.4" data-path="the-definite-integral.html"><a href="the-definite-integral.html#putting-it-all-together-30"><i class="fa fa-check"></i><b>5.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html"><i class="fa fa-check"></i><b>5.3</b> The Fundamental Theorem of Calculus</a>
<ul>
<li class="chapter" data-level="5.3.1" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#ftc-part-1-derivative-of-an-integral-ftc1"><i class="fa fa-check"></i><b>5.3.1</b> FTC Part 1: Derivative of an Integral (FTC1)</a></li>
<li class="chapter" data-level="5.3.2" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#ftc-part-2-evaluating-definite-integrals-ftc2"><i class="fa fa-check"></i><b>5.3.2</b> FTC Part 2: Evaluating Definite Integrals (FTC2)</a></li>
<li class="chapter" data-level="5.3.3" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#putting-it-all-together-31"><i class="fa fa-check"></i><b>5.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html"><i class="fa fa-check"></i><b>5.4</b> Indefinite Integrals and the Net Change Theorem</a>
<ul>
<li class="chapter" data-level="5.4.1" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#indefinite-integrals"><i class="fa fa-check"></i><b>5.4.1</b> Indefinite Integrals</a></li>
<li class="chapter" data-level="5.4.2" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#fundamental-connection-to-definite-integrals"><i class="fa fa-check"></i><b>5.4.2</b> Fundamental Connection to Definite Integrals</a></li>
<li class="chapter" data-level="5.4.3" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#the-net-change-theorem"><i class="fa fa-check"></i><b>5.4.3</b> The Net Change Theorem</a></li>
<li class="chapter" data-level="5.4.4" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#applications-of-net-change"><i class="fa fa-check"></i><b>5.4.4</b> Applications of Net Change</a></li>
<li class="chapter" data-level="5.4.5" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#putting-it-all-together-32"><i class="fa fa-check"></i><b>5.4.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.5" data-path="the-substitution-rule.html"><a href="the-substitution-rule.html"><i class="fa fa-check"></i><b>5.5</b> The Substitution Rule</a>
<ul>
<li class="chapter" data-level="5.5.1" data-path="the-substitution-rule.html"><a href="the-substitution-rule.html#putting-it-all-together-33"><i class="fa fa-check"></i><b>5.5.1</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="6" data-path="applications-of-integration.html"><a href="applications-of-integration.html"><i class="fa fa-check"></i><b>6</b> Applications of Integration</a>
<ul>
<li class="chapter" data-level="6.1" data-path="areas-between-curves.html"><a href="areas-between-curves.html"><i class="fa fa-check"></i><b>6.1</b> Areas Between Curves</a>
<ul>
<li class="chapter" data-level="6.1.1" data-path="areas-between-curves.html"><a href="areas-between-curves.html#when-curves-cross-absolute-value-form"><i class="fa fa-check"></i><b>6.1.1</b> When Curves Cross (Absolute Value Form)</a></li>
<li class="chapter" data-level="6.1.2" data-path="areas-between-curves.html"><a href="areas-between-curves.html#horizontal-slices-integrating-with-respect-to-y"><i class="fa fa-check"></i><b>6.1.2</b> Horizontal Slices (Integrating with Respect to <span class="math inline">\(y\)</span>)</a></li>
<li class="chapter" data-level="6.1.3" data-path="areas-between-curves.html"><a href="areas-between-curves.html#putting-it-all-together-34"><i class="fa fa-check"></i><b>6.1.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="volumes.html"><a href="volumes.html"><i class="fa fa-check"></i><b>6.2</b> Volumes</a>
<ul>
<li class="chapter" data-level="6.2.1" data-path="volumes.html"><a href="volumes.html#volume-of-a-cylinder"><i class="fa fa-check"></i><b>6.2.1</b> Volume of a Cylinder</a></li>
<li class="chapter" data-level="6.2.2" data-path="volumes.html"><a href="volumes.html#cross-sections-and-slicing"><i class="fa fa-check"></i><b>6.2.2</b> Cross-Sections and Slicing</a></li>
<li class="chapter" data-level="6.2.3" data-path="volumes.html"><a href="volumes.html#solids-of-revolution"><i class="fa fa-check"></i><b>6.2.3</b> Solids of Revolution</a></li>
<li class="chapter" data-level="6.2.4" data-path="volumes.html"><a href="volumes.html#pulling-it-all-together-1"><i class="fa fa-check"></i><b>6.2.4</b> Pulling It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="volumes-by-cylindrical-shells.html"><a href="volumes-by-cylindrical-shells.html"><i class="fa fa-check"></i><b>6.3</b> Volumes by Cylindrical Shells</a>
<ul>
<li class="chapter" data-level="6.3.1" data-path="volumes-by-cylindrical-shells.html"><a href="volumes-by-cylindrical-shells.html#pulling-it-all-together-2"><i class="fa fa-check"></i><b>6.3.1</b> Pulling It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.4" data-path="work.html"><a href="work.html"><i class="fa fa-check"></i><b>6.4</b> Work</a></li>
<li class="chapter" data-level="6.5" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html"><i class="fa fa-check"></i><b>6.5</b> Average Value of a Function</a>
<ul>
<li class="chapter" data-level="6.5.1" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#mean-value-theorem-for-integrals"><i class="fa fa-check"></i><b>6.5.1</b> Mean Value Theorem for Integrals</a></li>
<li class="chapter" data-level="6.5.2" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#application-average-velocity"><i class="fa fa-check"></i><b>6.5.2</b> Application: Average Velocity</a></li>
<li class="chapter" data-level="6.5.3" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#pulling-it-all-together-3"><i class="fa fa-check"></i><b>6.5.3</b> Pulling It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="7" data-path="techniques-of-integration.html"><a href="techniques-of-integration.html"><i class="fa fa-check"></i><b>7</b> Techniques of Integration</a>
<ul>
<li class="chapter" data-level="7.1" data-path="integration-by-parts.html"><a href="integration-by-parts.html"><i class="fa fa-check"></i><b>7.1</b> Integration by Parts</a>
<ul>
<li class="chapter" data-level="7.1.1" data-path="integration-by-parts.html"><a href="integration-by-parts.html#from-the-product-rule-to-integration-by-parts"><i class="fa fa-check"></i><b>7.1.1</b> From the Product Rule to Integration by Parts</a></li>
<li class="chapter" data-level="7.1.2" data-path="integration-by-parts.html"><a href="integration-by-parts.html#integration-by-parts-formula"><i class="fa fa-check"></i><b>7.1.2</b> Integration by Parts Formula</a></li>
<li class="chapter" data-level="7.1.3" data-path="integration-by-parts.html"><a href="integration-by-parts.html#choosing-u-and-dv"><i class="fa fa-check"></i><b>7.1.3</b> Choosing <span class="math inline">\(u\)</span> and <span class="math inline">\(dv\)</span></a></li>
<li class="chapter" data-level="7.1.4" data-path="integration-by-parts.html"><a href="integration-by-parts.html#putting-it-all-together-35"><i class="fa fa-check"></i><b>7.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.2" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html"><i class="fa fa-check"></i><b>7.2</b> Trigonometric Integrals</a>
<ul>
<li class="chapter" data-level="7.2.1" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#powers-of-sine-and-cosine"><i class="fa fa-check"></i><b>7.2.1</b> Powers of Sine and Cosine</a></li>
<li class="chapter" data-level="7.2.2" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#tangent-and-secant-integrals"><i class="fa fa-check"></i><b>7.2.2</b> Tangent and Secant Integrals</a></li>
<li class="chapter" data-level="7.2.3" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#special-integrals"><i class="fa fa-check"></i><b>7.2.3</b> Special Integrals</a></li>
<li class="chapter" data-level="7.2.4" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#putting-it-all-together-36"><i class="fa fa-check"></i><b>7.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html"><i class="fa fa-check"></i><b>7.3</b> Trigonometric Substitution</a>
<ul>
<li class="chapter" data-level="7.3.1" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html#inverse-substitution-idea"><i class="fa fa-check"></i><b>7.3.1</b> Inverse Substitution Idea</a></li>
<li class="chapter" data-level="7.3.2" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html#putting-it-all-together-37"><i class="fa fa-check"></i><b>7.3.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.4" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html"><i class="fa fa-check"></i><b>7.4</b> Integration of Rational Functions by Partial Fractions</a>
<ul>
<li class="chapter" data-level="7.4.1" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#general-strategy"><i class="fa fa-check"></i><b>7.4.1</b> General Strategy</a></li>
<li class="chapter" data-level="7.4.2" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#special-technique-rationalizing-substitutions"><i class="fa fa-check"></i><b>7.4.2</b> Special Technique: Rationalizing Substitutions</a></li>
<li class="chapter" data-level="7.4.3" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#putting-it-all-together-38"><i class="fa fa-check"></i><b>7.4.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.5" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html"><i class="fa fa-check"></i><b>7.5</b> Strategy for Integration</a>
<ul>
<li class="chapter" data-level="7.5.1" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#a-practical-strategy-for-integration"><i class="fa fa-check"></i><b>7.5.1</b> A Practical Strategy for Integration</a></li>
<li class="chapter" data-level="7.5.2" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#important-theoretical-insight"><i class="fa fa-check"></i><b>7.5.2</b> Important Theoretical Insight</a></li>
<li class="chapter" data-level="7.5.3" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#putting-it-all-together-39"><i class="fa fa-check"></i><b>7.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.6" data-path="integration-using-tables-and-computer-algebra-systems.html"><a href="integration-using-tables-and-computer-algebra-systems.html"><i class="fa fa-check"></i><b>7.6</b> Integration Using Tables and Computer Algebra Systems</a></li>
<li class="chapter" data-level="7.7" data-path="approximate-integration.html"><a href="approximate-integration.html"><i class="fa fa-check"></i><b>7.7</b> Approximate Integration</a>
<ul>
<li class="chapter" data-level="7.7.1" data-path="approximate-integration.html"><a href="approximate-integration.html#riemann-sum-approximations"><i class="fa fa-check"></i><b>7.7.1</b> Riemann Sum Approximations</a></li>
<li class="chapter" data-level="7.7.2" data-path="approximate-integration.html"><a href="approximate-integration.html#trapezoidal-rule"><i class="fa fa-check"></i><b>7.7.2</b> Trapezoidal Rule</a></li>
<li class="chapter" data-level="7.7.3" data-path="approximate-integration.html"><a href="approximate-integration.html#simpsons-rule"><i class="fa fa-check"></i><b>7.7.3</b> Simpson’s Rule</a></li>
<li class="chapter" data-level="7.7.4" data-path="approximate-integration.html"><a href="approximate-integration.html#error-behavior-conceptual"><i class="fa fa-check"></i><b>7.7.4</b> Error Behavior (Conceptual)</a></li>
<li class="chapter" data-level="7.7.5" data-path="approximate-integration.html"><a href="approximate-integration.html#putting-it-all-together-40"><i class="fa fa-check"></i><b>7.7.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.8" data-path="improper-integrals.html"><a href="improper-integrals.html"><i class="fa fa-check"></i><b>7.8</b> Improper Integrals</a>
<ul>
<li class="chapter" data-level="7.8.1" data-path="improper-integrals.html"><a href="improper-integrals.html#type-1-improper-integrals-infinite-intervals"><i class="fa fa-check"></i><b>7.8.1</b> Type 1 Improper Integrals — Infinite Intervals</a></li>
<li class="chapter" data-level="7.8.2" data-path="improper-integrals.html"><a href="improper-integrals.html#the-p-test-for-infinite-intervals"><i class="fa fa-check"></i><b>7.8.2</b> The <span class="math inline">\(p\)</span>-Test for Infinite Intervals</a></li>
<li class="chapter" data-level="7.8.3" data-path="improper-integrals.html"><a href="improper-integrals.html#type-2-improper-integrals-infinite-discontinuities"><i class="fa fa-check"></i><b>7.8.3</b> Type 2 Improper Integrals — Infinite Discontinuities</a></li>
<li class="chapter" data-level="7.8.4" data-path="improper-integrals.html"><a href="improper-integrals.html#comparison-test-for-improper-integrals"><i class="fa fa-check"></i><b>7.8.4</b> Comparison Test for Improper Integrals</a></li>
<li class="chapter" data-level="7.8.5" data-path="improper-integrals.html"><a href="improper-integrals.html#putting-it-all-together-41"><i class="fa fa-check"></i><b>7.8.5</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="8" data-path="further-applications-of-integration.html"><a href="further-applications-of-integration.html"><i class="fa fa-check"></i><b>8</b> Further Applications of Integration</a></li>
<li class="chapter" data-level="9" data-path="differential-equations.html"><a href="differential-equations.html"><i class="fa fa-check"></i><b>9</b> Differential Equations</a>
<ul>
<li class="chapter" data-level="9.1" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html"><i class="fa fa-check"></i><b>9.1</b> Modeling with Differential Equations</a>
<ul>
<li class="chapter" data-level="9.1.1" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#population-growth-models"><i class="fa fa-check"></i><b>9.1.1</b> Population Growth Models</a></li>
<li class="chapter" data-level="9.1.2" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#physical-model-motion-of-a-spring"><i class="fa fa-check"></i><b>9.1.2</b> Physical Model: Motion of a Spring</a></li>
<li class="chapter" data-level="9.1.3" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#general-differential-equations"><i class="fa fa-check"></i><b>9.1.3</b> General Differential Equations</a></li>
<li class="chapter" data-level="9.1.4" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#putting-it-all-together-42"><i class="fa fa-check"></i><b>9.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.2" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html"><i class="fa fa-check"></i><b>9.2</b> Direction Fields and Euler’s Method</a>
<ul>
<li class="chapter" data-level="9.2.1" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#direction-fields-slope-fields"><i class="fa fa-check"></i><b>9.2.1</b> Direction Fields (Slope Fields)</a></li>
<li class="chapter" data-level="9.2.2" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#autonomous-differential-equations"><i class="fa fa-check"></i><b>9.2.2</b> Autonomous Differential Equations</a></li>
<li class="chapter" data-level="9.2.3" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#eulers-method"><i class="fa fa-check"></i><b>9.2.3</b> Euler’s Method</a></li>
<li class="chapter" data-level="9.2.4" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#putting-it-all-together-43"><i class="fa fa-check"></i><b>9.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.3" data-path="separable-equations.html"><a href="separable-equations.html"><i class="fa fa-check"></i><b>9.3</b> Separable Equations</a>
<ul>
<li class="chapter" data-level="9.3.1" data-path="separable-equations.html"><a href="separable-equations.html#physical-application-example"><i class="fa fa-check"></i><b>9.3.1</b> Physical Application Example</a></li>
<li class="chapter" data-level="9.3.2" data-path="separable-equations.html"><a href="separable-equations.html#putting-it-all-together-44"><i class="fa fa-check"></i><b>9.3.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.4" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html"><i class="fa fa-check"></i><b>9.4</b> Models for Population Growth</a>
<ul>
<li class="chapter" data-level="9.4.1" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#the-law-of-natural-growth-exponential-model"><i class="fa fa-check"></i><b>9.4.1</b> The Law of Natural Growth (Exponential Model)</a></li>
<li class="chapter" data-level="9.4.2" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#the-logistic-growth-model"><i class="fa fa-check"></i><b>9.4.2</b> The Logistic Growth Model</a></li>
<li class="chapter" data-level="9.4.3" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#comparing-the-models"><i class="fa fa-check"></i><b>9.4.3</b> Comparing the Models</a></li>
<li class="chapter" data-level="9.4.4" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#putting-it-all-together-45"><i class="fa fa-check"></i><b>9.4.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.5" data-path="linear-equations.html"><a href="linear-equations.html"><i class="fa fa-check"></i><b>9.5</b> Linear Equations</a>
<ul>
<li class="chapter" data-level="9.5.1" data-path="linear-equations.html"><a href="linear-equations.html#the-integrating-factor-method"><i class="fa fa-check"></i><b>9.5.1</b> The Integrating Factor Method</a></li>
<li class="chapter" data-level="9.5.2" data-path="linear-equations.html"><a href="linear-equations.html#application-example-electric-circuits-rl-circuit"><i class="fa fa-check"></i><b>9.5.2</b> Application Example: Electric Circuits (RL Circuit)</a></li>
<li class="chapter" data-level="9.5.3" data-path="linear-equations.html"><a href="linear-equations.html#putting-it-all-together-46"><i class="fa fa-check"></i><b>9.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.6" data-path="predator-prey-systems.html"><a href="predator-prey-systems.html"><i class="fa fa-check"></i><b>9.6</b> Predator-Prey Systems</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="parametric-equations-and-polar-coordinates.html"><a href="parametric-equations-and-polar-coordinates.html"><i class="fa fa-check"></i><b>10</b> Parametric Equations and Polar Coordinates</a></li>
<li class="chapter" data-level="11" data-path="infinite-sequences-and-series.html"><a href="infinite-sequences-and-series.html"><i class="fa fa-check"></i><b>11</b> Infinite Sequences and Series</a>
<ul>
<li class="chapter" data-level="11.1" data-path="sequences.html"><a href="sequences.html"><i class="fa fa-check"></i><b>11.1</b> Sequences</a>
<ul>
<li class="chapter" data-level="11.1.1" data-path="sequences.html"><a href="sequences.html#limits-of-sequences"><i class="fa fa-check"></i><b>11.1.1</b> Limits of Sequences</a></li>
<li class="chapter" data-level="11.1.2" data-path="sequences.html"><a href="sequences.html#limit-laws-for-sequences"><i class="fa fa-check"></i><b>11.1.2</b> Limit Laws for Sequences</a></li>
<li class="chapter" data-level="11.1.3" data-path="sequences.html"><a href="sequences.html#some-additional-important-theorems"><i class="fa fa-check"></i><b>11.1.3</b> Some Additional Important Theorems</a></li>
<li class="chapter" data-level="11.1.4" data-path="sequences.html"><a href="sequences.html#special-sequences"><i class="fa fa-check"></i><b>11.1.4</b> Special Sequences</a></li>
<li class="chapter" data-level="11.1.5" data-path="sequences.html"><a href="sequences.html#putting-it-all-together-47"><i class="fa fa-check"></i><b>11.1.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.2" data-path="series.html"><a href="series.html"><i class="fa fa-check"></i><b>11.2</b> Series</a>
<ul>
<li class="chapter" data-level="11.2.1" data-path="series.html"><a href="series.html#sequences-vs.-series"><i class="fa fa-check"></i><b>11.2.1</b> Sequences vs. Series</a></li>
<li class="chapter" data-level="11.2.2" data-path="series.html"><a href="series.html#partial-sums-and-the-meaning-of-sum"><i class="fa fa-check"></i><b>11.2.2</b> Partial Sums and the Meaning of “Sum”</a></li>
<li class="chapter" data-level="11.2.3" data-path="series.html"><a href="series.html#geometric-series"><i class="fa fa-check"></i><b>11.2.3</b> Geometric Series</a></li>
<li class="chapter" data-level="11.2.4" data-path="series.html"><a href="series.html#telescoping-series"><i class="fa fa-check"></i><b>11.2.4</b> Telescoping Series</a></li>
<li class="chapter" data-level="11.2.5" data-path="series.html"><a href="series.html#fundamental-tests-and-theorems"><i class="fa fa-check"></i><b>11.2.5</b> Fundamental Tests and Theorems</a></li>
<li class="chapter" data-level="11.2.6" data-path="series.html"><a href="series.html#putting-it-all-together-48"><i class="fa fa-check"></i><b>11.2.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html"><i class="fa fa-check"></i><b>11.3</b> The Integral Test and Estimates of Sums</a>
<ul>
<li class="chapter" data-level="11.3.1" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#the-p-series"><i class="fa fa-check"></i><b>11.3.1</b> The p-Series</a></li>
<li class="chapter" data-level="11.3.2" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#error-estimates"><i class="fa fa-check"></i><b>11.3.2</b> Error Estimates</a></li>
<li class="chapter" data-level="11.3.3" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#putting-it-all-together-49"><i class="fa fa-check"></i><b>11.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.4" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html"><i class="fa fa-check"></i><b>11.4</b> The Comparison Tests</a>
<ul>
<li class="chapter" data-level="11.4.1" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#the-comparison-test"><i class="fa fa-check"></i><b>11.4.1</b> The Comparison Test</a></li>
<li class="chapter" data-level="11.4.2" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#the-limit-comparison-test"><i class="fa fa-check"></i><b>11.4.2</b> The Limit Comparison Test</a></li>
<li class="chapter" data-level="11.4.3" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#estimating-sums-using-comparison"><i class="fa fa-check"></i><b>11.4.3</b> Estimating Sums Using Comparison</a></li>
<li class="chapter" data-level="11.4.4" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#putting-it-all-together-50"><i class="fa fa-check"></i><b>11.4.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.5" data-path="alternating-series.html"><a href="alternating-series.html"><i class="fa fa-check"></i><b>11.5</b> Alternating Series</a>
<ul>
<li class="chapter" data-level="11.5.1" data-path="alternating-series.html"><a href="alternating-series.html#alternating-series-test-ast"><i class="fa fa-check"></i><b>11.5.1</b> Alternating Series Test (AST)</a></li>
<li class="chapter" data-level="11.5.2" data-path="alternating-series.html"><a href="alternating-series.html#estimating-sums-of-alternating-series"><i class="fa fa-check"></i><b>11.5.2</b> Estimating Sums of Alternating Series</a></li>
<li class="chapter" data-level="11.5.3" data-path="alternating-series.html"><a href="alternating-series.html#putting-it-all-together-51"><i class="fa fa-check"></i><b>11.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.6" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html"><i class="fa fa-check"></i><b>11.6</b> Absolute Convergence and the Ratio and Root Tests</a></li>
<li class="chapter" data-level="11.7" data-path="absolute-convergence.html"><a href="absolute-convergence.html"><i class="fa fa-check"></i><b>11.7</b> Absolute Convergence</a>
<ul>
<li class="chapter" data-level="11.7.1" data-path="absolute-convergence.html"><a href="absolute-convergence.html#the-ratio-test"><i class="fa fa-check"></i><b>11.7.1</b> The Ratio Test</a></li>
<li class="chapter" data-level="11.7.2" data-path="absolute-convergence.html"><a href="absolute-convergence.html#the-root-test"><i class="fa fa-check"></i><b>11.7.2</b> The Root Test</a></li>
<li class="chapter" data-level="11.7.3" data-path="absolute-convergence.html"><a href="absolute-convergence.html#rearrangements-of-series"><i class="fa fa-check"></i><b>11.7.3</b> Rearrangements of Series</a></li>
<li class="chapter" data-level="11.7.4" data-path="absolute-convergence.html"><a href="absolute-convergence.html#putting-it-all-together-52"><i class="fa fa-check"></i><b>11.7.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.8" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html"><i class="fa fa-check"></i><b>11.8</b> Strategy for Testing Series</a>
<ul>
<li class="chapter" data-level="11.8.1" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html#worked-strategy-examples"><i class="fa fa-check"></i><b>11.8.1</b> Worked Strategy Examples</a></li>
<li class="chapter" data-level="11.8.2" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html#putting-it-all-together-53"><i class="fa fa-check"></i><b>11.8.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.9" data-path="power-series.html"><a href="power-series.html"><i class="fa fa-check"></i><b>11.9</b> Power Series</a>
<ul>
<li class="chapter" data-level="11.9.1" data-path="power-series.html"><a href="power-series.html#power-series-as-functions"><i class="fa fa-check"></i><b>11.9.1</b> Power Series as Functions</a></li>
<li class="chapter" data-level="11.9.2" data-path="power-series.html"><a href="power-series.html#radius-and-interval-of-convergence"><i class="fa fa-check"></i><b>11.9.2</b> Radius and Interval of Convergence</a></li>
<li class="chapter" data-level="11.9.3" data-path="power-series.html"><a href="power-series.html#putting-it-all-together-54"><i class="fa fa-check"></i><b>11.9.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.10" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html"><i class="fa fa-check"></i><b>11.10</b> Representations of Functions as Power Series</a>
<ul>
<li class="chapter" data-level="11.10.1" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html#the-fundamental-geometric-series-model"><i class="fa fa-check"></i><b>11.10.1</b> The Fundamental Geometric Series Model</a></li>
</ul></li>
</ul></li>
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<div id="absolute-convergence" class="section level2 hasAnchor" number="11.7">
<h2><span class="header-section-number">11.7</span> Absolute Convergence<a href="absolute-convergence.html#absolute-convergence" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div class="definition">
<p><span id="def:unlabeled-div-49" class="definition"><strong>Definition 11.17  </strong></span>Given a series
<span class="math display">\[
\sum_{n=1}^{\infty} a_n,
\]</span>
consider the associated series of absolute values:
<span class="math display">\[
\sum_{n=1}^{\infty} |a_n|.
\]</span></p>
<p>A series is called:</p>
<ul>
<li><p><strong>Absolutely convergent</strong> if<br />
<span class="math display">\[
\sum |a_n| \quad \text{converges}
\]</span></p></li>
<li><p><strong>Conditionally convergent</strong> if<br />
<span class="math display">\[
\sum a_n \text{ converges but } \sum |a_n| \text{ diverges}
\]</span></p></li>
</ul>
</div>
<div class="theorem">
<p><span id="thm:unlabeled-div-50" class="theorem"><strong>Theorem 11.12  </strong></span><strong>Absolute Convergence Theorem</strong></p>
<p>If a series is <strong>absolutely convergent</strong>, then it is <strong>convergent</strong>.</p>
</div>
<p><span class="math display">\[
\sum |a_n| \text{ convergent } \;\Rightarrow\; \sum a_n \text{ convergent}
\]</span></p>
<p>This means:</p>
<blockquote>
<p>Absolute convergence is <strong>stronger</strong> than ordinary convergence.</p>
</blockquote>
<div class="example">
<p><span id="exm:unlabeled-div-51" class="example"><strong>Example 11.22  </strong></span>Absolute Convergence</p>
<p>Determine if the series is convergent
<span class="math display">\[
\sum_{n=1}^{\infty} (-1)^{n-1}\frac{1}{n^2}
\]</span></p>
<p><strong>Solution:</strong>
<span class="math display">\[
\sum |a_n| = \sum_{n=1}^{\infty} \frac{1}{n^2}
\]</span>
This is a <strong>p-series</strong> with <span class="math inline">\(p=2&gt;1\)</span>, so it converges. The series is <strong>absolutely convergent</strong>, and therefore <strong>convergent</strong>.</p>
</div>
<div class="example">
<p><span id="exm:unlabeled-div-52" class="example"><strong>Example 11.23  </strong></span>Conditional Convergence</p>
<p>Determine if the series is convergent
<span class="math display">\[
\sum_{n=1}^{\infty} (-1)^{n-1}\frac{1}{n}
\quad \text{(alternating harmonic series)}
\]</span></p>
<p><strong>Solution:</strong></p>
<p>This series converges by the <strong>Alternating Series Test</strong>, but</p>
<p><span class="math display">\[
\sum |a_n| = \sum \frac{1}{n}
\]</span>
is the harmonic series, which diverges.</p>
<p>So, the conclusion is that the series is <strong>conditionally convergent</strong>.</p>
</div>
<div class="example">
<p><span id="exm:unlabeled-div-53" class="example"><strong>Example 11.24  </strong></span>Irregular Signs</p>
<p>Consider
<span class="math display">\[
\sum_{n=1}^{\infty} \frac{\cos n}{n^2}
\]</span></p>
<p>This is <strong>not alternating</strong> (signs are irregular), but:
<span class="math display">\[
\sum |a_n| = \sum \frac{|\cos n|}{n^2}
\le \sum \frac{1}{n^2}.
\]</span>
Since <span class="math inline">\(\sum \frac{1}{n^2}\)</span> converges, the Comparison Test gives convergence.</p>
<p>Conclusion: The series is <strong>absolutely convergent</strong>, hence <strong>convergent</strong>.</p>
</div>
<hr />
<div id="the-ratio-test" class="section level3 hasAnchor" number="11.7.1">
<h3><span class="header-section-number">11.7.1</span> The Ratio Test<a href="absolute-convergence.html#the-ratio-test" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="theorem">
<p><span id="thm:unlabeled-div-54" class="theorem"><strong>Theorem 11.13  </strong></span><strong>Ratio Test</strong></p>
<p>For a series <span class="math inline">\(\sum a_n\)</span>, define
<span class="math display">\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.
\]</span></p>
<ul>
<li>If <span class="math inline">\(L &lt; 1\)</span> → <strong>absolutely convergent</strong><br />
</li>
<li>If <span class="math inline">\(L &gt; 1\)</span> or <span class="math inline">\(L = \infty\)</span> → <strong>divergent</strong><br />
</li>
<li>If <span class="math inline">\(L = 1\)</span> → <strong>inconclusive</strong><br />
</li>
</ul>
</div>
<div class="example">
<p><span id="exm:unlabeled-div-55" class="example"><strong>Example 11.25  </strong></span>Ratio Test</p>
<p>Determine if the series converges
<span class="math display">\[
\sum_{n=1}^{\infty} (-1)^n \frac{n^3}{3^n}
\]</span></p>
<p><strong>Solution:</strong></p>
<p><span class="math display">\[
\left|\frac{a_{n+1}}{a_n}\right|
= \frac{(n+1)^3}{3^{n+1}} \cdot \frac{3^n}{n^3}
= \frac{1}{3}\left(\frac{n+1}{n}\right)^3
\to \frac{1}{3}
\]</span></p>
<p>Conclusion:
Since <span class="math inline">\(L=\frac{1}{3}&lt;1\)</span>, the series is <strong>absolutely convergent</strong>.</p>
</div>
<hr />
</div>
<div id="the-root-test" class="section level3 hasAnchor" number="11.7.2">
<h3><span class="header-section-number">11.7.2</span> The Root Test<a href="absolute-convergence.html#the-root-test" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="theorem">
<p><span id="thm:unlabeled-div-56" class="theorem"><strong>Theorem 11.14  </strong></span><strong>The Root Test</strong></p>
<p>For a series <span class="math inline">\(\sum a_n\)</span>, define
<span class="math display">\[
L = \lim_{n \to \infty} \sqrt[n]{|a_n|}.
\]</span></p>
<ul>
<li>If <span class="math inline">\(L &lt; 1\)</span> → <strong>absolutely convergent</strong><br />
</li>
<li>If <span class="math inline">\(L &gt; 1\)</span> → <strong>divergent</strong><br />
</li>
<li>If <span class="math inline">\(L = 1\)</span> → <strong>inconclusive</strong><br />
</li>
</ul>
</div>
<div class="example">
<p><span id="exm:unlabeled-div-57" class="example"><strong>Example 11.26  </strong></span>Root Test</p>
<p>Determine if the series converges
<span class="math display">\[
\sum_{n=1}^{\infty} \left(\frac{2n+3}{3n+2}\right)^n
\]</span></p>
<p><strong>Solution:</strong>
<span class="math display">\[
\sqrt[n]{|a_n|} = \frac{2n+3}{3n+2} \to \frac{2}{3}
\]</span></p>
<p>Conclusion:
Since <span class="math inline">\(\frac{2}{3}&lt;1\)</span>, the series is <strong>absolutely convergent</strong>.</p>
</div>
<hr />
</div>
<div id="rearrangements-of-series" class="section level3 hasAnchor" number="11.7.3">
<h3><span class="header-section-number">11.7.3</span> Rearrangements of Series<a href="absolute-convergence.html#rearrangements-of-series" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><strong>Finite vs Infinite Sums</strong></p>
<ul>
<li>Finite sums → rearranging terms <strong>does not change the sum</strong><br />
</li>
<li>Infinite series → rearranging terms <strong>can change the sum</strong></li>
</ul>
<p><strong>Fundamental Results</strong></p>
<blockquote>
<p>Absolutely Convergent Series: Any rearrangement has the <strong>same sum</strong></p>
</blockquote>
<blockquote>
<p>Conditionally Convergent Series: Rearrangements can change the sum.</p>
</blockquote>
<p>In fact, conditionally convergent series can be rearranged to converge to <strong>any real number</strong>. (This is known as <strong>Riemann’s Rearrangement Theorem</strong>.)</p>
<p>Absolute convergence behaves like finite sums. Conditional convergence behaves like order-sensitive infinite processes.</p>
<hr />
</div>
<div id="putting-it-all-together-52" class="section level3 hasAnchor" number="11.7.4">
<h3><span class="header-section-number">11.7.4</span> Putting It All Together<a href="absolute-convergence.html#putting-it-all-together-52" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="conceptual-takeaways-56" class="section level4 hasAnchor" number="11.7.4.1">
<h4><span class="header-section-number">11.7.4.1</span> Conceptual Takeaways<a href="absolute-convergence.html#conceptual-takeaways-56" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Absolute convergence guarantees <strong>stability</strong><br />
</li>
<li>Conditional convergence means <strong>cancellation-based convergence</strong><br />
</li>
<li>Irregular sign series require <strong>absolute value analysis</strong><br />
</li>
<li>Ratio &amp; Root Tests detect <strong>exponential/factorial growth</strong><br />
</li>
<li>Rearrangements reveal the <strong>structure</strong> of infinite sums<br />
</li>
<li>Absolute convergence = <strong>safe convergence</strong><br />
</li>
<li>Conditional convergence = <strong>fragile convergence</strong></li>
</ul>
<hr />
</div>
<div id="skills-you-should-be-able-to-do-54" class="section level4 hasAnchor" number="11.7.4.2">
<h4><span class="header-section-number">11.7.4.2</span> Skills You Should Be Able to Do<a href="absolute-convergence.html#skills-you-should-be-able-to-do-54" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Recognize absolute vs conditional convergence<br />
</li>
<li>Form the absolute-value series <span class="math inline">\(\sum |a_n|\)</span><br />
</li>
<li>Detect irregular sign patterns<br />
</li>
<li>Apply Comparison Test to <span class="math inline">\(\sum |a_n|\)</span><br />
</li>
<li>Use Alternating Series Test vs absolute convergence<br />
</li>
<li>Apply Ratio Test correctly<br />
</li>
<li>Apply Root Test correctly<br />
</li>
<li>Recognize inconclusive cases<br />
</li>
<li>Distinguish absolute vs conditional convergence<br />
</li>
<li>Understand stability of sums<br />
</li>
<li>Interpret rearrangements<br />
</li>
<li>Classify convergence types<br />
</li>
<li>Choose correct test based on structure:
<ul>
<li>factorials → Ratio Test<br />
</li>
<li>exponentials → Ratio Test<br />
</li>
<li>powers → Root Test<br />
</li>
<li>irregular signs → absolute convergence<br />
</li>
<li>alternating → AST<br />
</li>
<li>rational functions → Comparison / Limit Comparison</li>
</ul></li>
</ul>
<hr />
</div>
<div id="problems-58" class="section level4 hasAnchor" number="11.7.4.3">
<h4><span class="header-section-number">11.7.4.3</span> Problems<a href="absolute-convergence.html#problems-58" class="anchor-section" aria-label="Anchor link to header"></a></h4>
</div>
</div>
</div>
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