Chapter 7: Q. 12 (page 652)

Explain why you must use two convergence tests to show that a series $\sum_{k = 1}^{\infty} a_{k} Converges conditionally$

Short Answer

Expert verified

$For conditionally convergent series \sum_{k = 1}^{\infty} |a_{k}| must diverge and \sum_{k = 1}^{\infty} a_{k} converges .$

Step by step solution

Step 1. Given

$\sum_{k = 1}^{\infty} a_{k} Converges conditionally$ .

Step 2. Explanation

$The series \sum_{k = 1}^{\infty} a_{k} will be converge conditionally if \sum_{k = 1}^{\infty} |a_{k}| Diverges . Hence, for conditionally convergent series \sum_{k = 1}^{\infty} |a_{k}| must diverge and \sum_{k = 1}^{\infty} a_{k} converges . Hence, for conditionally convergent series two test are required .$

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Explain why you must use two convergence tests to show that a series $\sum_{k = 1}^{\infty} a_{k} Converges conditionally$

Short Answer

Step by step solution

Step 1. Given

Step 2. Explanation

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Explain why you must use two convergence tests to show that a series∑k=1∞akConvergesconditionally

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Step by step solution

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Explain why you must use two convergence tests to show that a series $\sum_{k = 1}^{\infty} a_{k} Converges conditionally$