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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett
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Absolute and conditional convergence Determine

Chapter 10, Problem 40E

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QUESTION:

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally.

\(\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}\)

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QUESTION:

39-46. Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally.

\(\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}\)

ANSWER:

Problem 40E

Absolute and conditional convergence Determine whether the following series converge absolutely or conditionally.

Solution

Step 1

In this problem we have to find whether the seriesconverges absolutely or conditionally.

Absolute convergence: is absolutely convergent if is convergent.

Conditional convergence: is conditionally convergent if is divergent and is convergent.

Let us check the convergence of the series by alternating series test.

Alternating series test:

Suppose that for there exists a N so that for all

  1. is positive and decreasing
  2. .

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