Strategies for Testing Series

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The Fundamental Theorem of Calculus

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The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
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The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
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Substitution

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Integrals of Trig Functions

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Partial Fractions

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Strategies of Integration

Substitution
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Improper Integrals

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Infinite Series

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Integral Test

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Comparison Tests

The Basic Comparison Test
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Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence
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The Ratio and Root Tests

The Ratio Test
The Root Test
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Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
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Power Series Centered at $x=a$

Representing Functions as Power Series

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Taylor and Maclaurin Series

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Taylor Polynomials
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Partial Derivatives

Visualizing Functions in 3 Dimensions
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Multiple Integrals

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What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
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Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

Examples, Part 2

DO: Test the following series for convergence, divergence, conditional convergence and absolute convergence, when possible.

There are more examples to work on the previous page, which you might want to work on also before checking any answers.

$\displaystyle\sum_n \frac{(-1)^n}{\ln(n)}$;

$\displaystyle\sum_n \frac{(-1)^n(n+1)}{2n}$;

$\displaystyle\sum_n \frac{\cos(n)}{n^2}$;

$\displaystyle\sum_n n^2\left(\frac23\right)^n$;

$\displaystyle\sum_n \frac{(3/2)^n}{n^5}$; and

$\displaystyle\sum_n \left ( 1 - \frac{1}{n^2} \right )^{n^3}$.

The video below will discuss the following series. We have put the written results below the video.

$\displaystyle\sum_n \frac{(-1)^n}{\ln(n)}$ converges by the Alternating Series Test; the convergence is conditional by a comparison test.

$\displaystyle\sum_n \frac{(-1)^n(n+1)}{2n}$ is an alternating series, but it diverges, not by the AST (which does not indicate divergence) but by the Divergence Test.

$\displaystyle\sum_n \frac{\cos(n)}{n^2}$ converges absolutely, by comparison to $\displaystyle\sum \frac{1}{n^2}$;

$\displaystyle\sum_n n^2\left(\frac23\right)^n$ converges absolutely by the Ratio Test;

$\displaystyle\sum_n \frac{(3/2)^n}{n^5}$ diverges by the Ratio Test;

$\displaystyle\sum_n \left ( 1 - \frac{1}{n^2} \right )^{n^3}$ converges absolutely by the Root Test.