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Chapter 7: Q. 9 (page 652)

What condition(s) must a series $\sum_{k = 1}^{\infty} a_{k}$ satisfy in order for the series to be absolutely convergent?

Short Answer

Expert verified

$The series \sum_{k = 1}^{\infty} a_{k} will be converge absolutely if \sum_{k = 1}^{\infty} |a_{k}| converges . Hence, for Conditionally convergent series \sum_{k = 1}^{\infty} a_{k} must converges and \sum_{k = 1}^{\infty} a_{k} converges .$

Step by step solution

01

Step 1. Given

The given series is $\sum_{k = 1}^{\infty} a_{k}$

02

Step 2. Test for absolutely convergence

$The series \sum_{k = 1}^{\infty} a_{k} will be converge absolutely if \sum_{k = 1}^{\infty} |a_{k}| converges . Hence, for Conditionally convergent series \sum_{k = 1}^{\infty} a_{k} must converges and \sum_{k = 1}^{\infty} a_{k} converges .$

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