Chapter 7: Q.1 c) (page 630)

if $0 \leq a_{k} \leq \frac{1}{k}$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} a_{k}$ converges. The objective is to whether determine the statement is true or false

Short Answer

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false

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To determine whether given statement is true or false

To determine whether the given statement is true or not by using comparison test

According to comparison test

$\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ are two series with positive terms such that $0 \leq a_{k} \leq b_{k}$ for every positive integer k

If $\sum_{k = 1}^{\infty} b_{k}$ converges then the series $\sum_{k = 1}^{\infty} a_{k}$ also converges

If the series $\sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ is divergent by the power series, So $\sum_{k = 1}^{\infty} a_{k}$ also divergence

Hence the given statement is wrong.

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if $0 \leq a_{k} \leq \frac{1}{k}$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} a_{k}$ converges. The objective is to whether determine the statement is true or false

Short Answer

Step by step solution

To determine whether given statement is true or false

According to comparison test

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if 0≤ak≤1kfor every positive integer k, then the series ∑k=1∞akconverges. The objective is to whether determine the statement is true or false

Short Answer

Step by step solution

To determine whether given statement is true or false

According to comparison test

One App. One Place for Learning.

Most popular questions from this chapter

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if $0 \leq a_{k} \leq \frac{1}{k}$ for every positive integer k, then the series $\sum_{k = 1}^{\infty} a_{k}$ converges. The objective is to whether determine the statement is true or false