Chapter 7: Q. 48 (page 653)

Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$

Short Answer

Expert verified

The series is absolutely convergent.

Step by step solution

Step 1. Given information.

Consider the given question,

$\sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$

Step 2. Consider the general series.

The general term of the series $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$ is given below,

$a_{k} = \frac{\sin \sqrt{k}}{k^{3 / 2}}$

The limit comparison test states that for localid="1649154816940" $\sum_{k = 1}^{\infty} a_{k}, \sum_{k = 1}^{\infty} b_{k}$ be two series with positive terms such that $0 \leq a_{k} \leq b_{k}$ for every positive integer k. If the series $\sum_{k = 1}^{\infty} b_{k}$ converges, then the series $\sum_{k = 1}^{\infty} a_{k}$ converges.

The terms of the series $\sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$ is positive.

Step 3. Consider the series ∑k=1∞ bk.

The expression $\frac{\sin \sqrt{k}}{k^{3 / 2}}$ satisfies $|\frac{\sin \sqrt{k}}{k^{3 / 2}}| \leq \frac{1}{k^{3 / 2}}$ .

The series $\sum_{k = 1}^{\infty} b_{k}$ is given below,

$\sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{3 / 2}}$

The above series is convergent and converges absolutely.

Hence, the given series is absolutely convergent.

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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the general series.

Step 3. Consider the series ∑k=1∞ bk.

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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.∑k=1∞sinkk3/2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the general series.

Step 3. Consider the series ∑k=1∞ bk.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} \frac{\sin \sqrt{k}}{k^{3 / 2}}$