Chapter 7: Q. 17 (page 652)

Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series $\sum_{k = 1}^{\infty} a_{2 k - 1}$ and $\sum_{k = 1}^{\infty} a_{2 k}$ both converge absolutely, the series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely.

Short Answer

Expert verified

As the given series is absolutely convergent and $\sum_{k = 1}^{\infty} a_{2 k - 1}$ is the series of odd numbers and $\sum_{k = 1}^{\infty} a_{2 k}$ is the series of even numbers.

Step by step solution

. Given information.

Consider the given question,

$\{a_{k}\}$ is the sequence of positive numbers.

Step 2. Explain the series.

The series $\sum_{k = 1}^{\infty} a_{2 k - 1}$ is the series of odd numbers $\sum_{k = 1}^{\infty} a_{2 k}$ is the series of even numbers. Both the series converge absolutely.

The series $\sum_{k = 1}^{\infty} a_{k}$ is the sum of series $\sum_{k = 1}^{\infty} a_{2 k - 1}, \sum_{k = 1}^{\infty} a_{2 k}$ .

As the sum of two convergent series is convergent.

Moreover, the series $\sum_{k = 1}^{\infty} a_{2 k - 1}, \sum_{k = 1}^{\infty} a_{2 k}$ is absolutely convergent.

Therefore, the series $\sum_{k = 1}^{\infty} a_{k}$ is convergent.

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Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series $\sum_{k = 1}^{\infty} a_{2 k - 1}$ and $\sum_{k = 1}^{\infty} a_{2 k}$ both converge absolutely, the series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely.

Short Answer

Step by step solution

. Given information.

Step 2. Explain the series.

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Letakbe a sequence of positive numbers. Explain why, if the series∑k=1∞a2k-1and∑k=1∞a2kboth converge absolutely, the series∑k=1∞akconverges absolutely.

Short Answer

Step by step solution

. Given information.

Step 2. Explain the series.

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Most popular questions from this chapter

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Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series $\sum_{k = 1}^{\infty} a_{2 k - 1}$ and $\sum_{k = 1}^{\infty} a_{2 k}$ both converge absolutely, the series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely.