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Chapter 7: Q. 3 (page 652)

Explain the term alternating series.

Short Answer

Expert verified

$\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} . . . . . . . . . .$

Step by step solution

01

Step 1. To explain

The term alternating series.

02

Step 2. Explanation

$Let a_{k} be the sequence of positive integers . The alternating series is in the form \sum_{k = 1}^{\infty} {(- 1)}^{k - 1} a_{k} or \sum_{k = 1}^{\infty} {(- 1)}^{k} a_{k} Consider the series \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1}}{k} Expanding \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} . . . . . . . . . . Hence the series is alternating series .$

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