Q. 42 Use any convergence test from Se... [FREE SOLUTION]

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Chapter 7: Q. 42 (page 640)

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.
${鈭�}_{k = 1}^{鈭�} \frac{3}{k (k + 2)}$

Short Answer

Expert verified

The given series converges.

Step by step solution

Step 1. Given Information.

The given series is ${鈭�}_{k = 1}^{鈭�} \frac{3}{k (k + 2)} .$

Step 2. Determine whether the series converges or diverges.

To determine whether the series converges or diverges we will use the comparison test since the series has positive terms that meet the hypothesis of the test.

Let ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k (k + 2)} and {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} .$

So, $0 鈮� \frac{1}{k (k + 2)} 鈮� \frac{1}{k^{2}} .$

Now, ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} is of the form {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{p}} .$

If p > 1 then it is converges or ifp < 1then it is diverges.

Here, $p = 2 > 1$ thus, ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$ converges.

By the comparison test, ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k (k + 2)}$ also converges.

Thus, $3 {鈭�}_{k = 1}^{鈭�} a_{k} = 3 {鈭�}_{k = 1}^{鈭�} \frac{1}{k (k + 2)} = {鈭�}_{k = 1}^{鈭�} \frac{3}{k (k + 2)}$ converges.

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91影视

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.
${鈭�}_{k = 1}^{鈭�} \frac{3}{k (k + 2)}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the series converges or diverges.

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91影视

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.鈭�k=1鈭�3kk+2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the series converges or diverges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.
${鈭�}_{k = 1}^{鈭�} \frac{3}{k (k + 2)}$