Chapter 7: Q. 54 (page 632)

Prove that if $\sum_{k = 1}^{\infty} a_{k}$ is a convergent series with $a_{k} \geq 0$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} \frac{a_{k}}{k}$ converges.

Short Answer

Expert verified

Series $\sum_{k = 1}^{\infty} a_{k}$ is a convergent where $a_{k} \geq 0$ for every positive integer k.

so $N > 0$ will also exist such that $a_{k} < 1$ for all $n > N .$

when $n > N & a_{k} < 1,$ then $0 \leq \frac{a_{k}^{}}{k} \leq a_{k} .$

According to The Comparison Test, if role="math" localid="1649139323101" $0 \leq \frac{a_{k}^{}}{k} \leq a_{k} & \sum_{k = 1}^{\infty} a_{k}$ converges then series $\sum_{k = 1}^{\infty} \frac{a_{k}^{}}{k}$ also converges.

Step by step solution

Step 1. Given Information.

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

Series $\sum_{k = 1}^{\infty} a_{k}$ is a convergent where $a_{k} \geq 1$ for every positive integerk.

Step 2. Proof.

Series $\sum_{k = 1}^{\infty} a_{k}$ is a convergent where $a_{k} \geq 0$ for every positive integer k.

so $N > 0$ will also exist such that $a_{k} < 1$ for all $n > N .$

when $n > N & a_{k} < 1,$ thenrole="math" localid="1649139290641" $0 \leq \frac{a_{k}^{}}{k} \leq a_{k} .$

According to The Comparison Test, if role="math" localid="1649139270963" $0 \leq \frac{a_{k}^{}}{k} \leq a_{k} & \sum_{k = 1}^{\infty} a_{k}$ converges then series $\sum_{k = 1}^{\infty} \frac{a_{k}^{}}{k}$ also converges.

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Prove that if $\sum_{k = 1}^{\infty} a_{k}$ is a convergent series with $a_{k} \geq 0$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} \frac{a_{k}}{k}$ converges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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Prove that if ∑k=1∞akis a convergent series with ak≥0for every positive integer k, then the series ∑k=1∞akkconverges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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Prove that if $\sum_{k = 1}^{\infty} a_{k}$ is a convergent series with $a_{k} \geq 0$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} \frac{a_{k}}{k}$ converges.