Q. 42 Use any convergence tests from S... [FREE SOLUTION]

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Chapter 7: Q. 42 (page 653)

Use any convergence tests from Sections 7.4鈥�7.7 to determine whether the series in Exercises 36鈥�51 converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \sqrt{k}$ ³+3k²-5k²+3

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{\sqrt{k^{3} + 3 k^{2} - 5}}{k^{2} + 3}$ converges conditionally.

Step by step solution

Step 1. Given Information.

The series:

${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{\sqrt{k^{3} + 3 k^{2} - 5}}{k^{2} + 3}$

Step 2. Consider the series.

Consider the series,

${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{\sqrt{k^{3} + 3 k^{2} - 5}}{k^{2} + 3} = {鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} a_{k} a n d a_{k}$ is of positive terms.

The series is decreasing as $a_{k + 1} < a_{k}$ .

$\underset{k 鈫� 鈭�}{l i m} a_{k} = \underset{k 鈫� 鈭�}{l i m} \frac{\sqrt{k^{3} + 3 k^{2} - 5}}{k^{2} + 3} = 0$

By Alternating test series, the series is convergent.

Step 3. rewrite the series.

The dominant form of the series is,

$\underset{k 鈫� 鈭�}{l i m} b_{k} = \underset{k 鈫� 鈭�}{l i m} \frac{k^{\frac{3}{2}}}{k^{2}} = \underset{k 鈫� 鈭�}{l i m} \frac{1}{k^{\frac{3}{2}}}$

Step 4. Find limk→∞akbk.

$\underset{k 鈫� 鈭�}{l i m} \frac{a_{k}}{b_{k}} = \underset{k 鈫� 鈭�}{l i m} \frac{\frac{\sqrt{k^{3} + 3 k^{2} - 5}}{k^{2} + 3}}{\frac{1}{k^{\frac{3}{2}}}} = \underset{k 鈫� 鈭�}{l i m} \frac{k^{2} \sqrt{1 + \frac{3 k^{2}}{k^{3}} - \frac{5}{k^{3}}}}{k^{2} (1 + \frac{3}{k^{3}})} = 1$

So $\underset{k 鈫� 鈭�}{l i m} b_{k}$ is divergent and so $\underset{k 鈫� 鈭�}{l i m} a_{k}$ is also divergent.

So the given series is conditionally convergent.

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91影视

Use any convergence tests from Sections 7.4鈥�7.7 to determine whether the series in Exercises 36鈥�51 converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \sqrt{k}$ ³+3k²-5k²+3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Consider the series.

Step 3. rewrite the series.

Step 4. Find limk→∞akbk.

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91影视

Use any convergence tests from Sections 7.4鈥�7.7 to determine whether the series in Exercises 36鈥�51 converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.鈭�k=1鈭�(-1)kk3+3k2-5k2+3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Consider the series.

Step 3. rewrite the series.

Step 4. Find limk→∞akbk.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

Use any convergence tests from Sections 7.4鈥�7.7 to determine whether the series in Exercises 36鈥�51 converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \sqrt{k}$ ³+3k²-5k²+3