Q. 49 Use any convergence tests to det... [FREE SOLUTION]

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Chapter 7: Q. 49 (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.
${鈭�}_{k = 1}^{鈭�} \frac{\cos k}{k^{2}}$

Short Answer

Expert verified

The series converges absolutely.

Step by step solution

Step 1. Given information.

Consider the given question,

${鈭�}_{k = 1}^{鈭�} \frac{\cos k}{k^{2}}$

Step 2. Consider the general series.

The general term of the series ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{\cos k}{k^{2}}$ is given below,

$a_{k} = \frac{\cos k}{k^{2}}$

The limit comparison test states that for ${鈭�}_{k = 1}^{鈭�} a_{k}, {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms such that $0 鈮� a_{k} 鈮� b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

Step 3. Consider the term of the given series as positive.

The given expression satisfies $|\frac{\cos k}{k^{2}}| 鈮� \frac{1}{k^{2}}$ .

The series ${鈭�}_{k = 1}^{鈭�} b_{k}$ for the given series isrole="math" localid="1649155585096" ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$ .

The series ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$ is convergent by p-series test.

The above series is convergent and converges absolutely.

Hence, the given series is absolutely convergent.

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91影视

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.
${鈭�}_{k = 1}^{鈭�} \frac{\cos k}{k^{2}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the general series.

Step 3. Consider the term of the given series as positive.

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91影视

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.鈭�k=1鈭�coskk2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the general series.

Step 3. Consider the term of the given series as positive.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Calculus

Probability and Statistics

Discrete Mathematics

Geometry

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.
${鈭�}_{k = 1}^{鈭�} \frac{\cos k}{k^{2}}$