Q. 48 Use any convergence test from th... [FREE SOLUTION]

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Chapter 7: Q. 48 (page 632)

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$ is convergent.

Step by step solution

Step 1. Given information.

The given series is the following.

${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$

Step 2. The Limit Comparison Test.

Consider a series ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{\frac{3}{2}}}$ by taking the dominant term of numerator and denominator of ${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3} .$

Find the value of $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} .$

$\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{\sqrt{k}}{k^{2} + 3}}{\frac{1}{k^{\frac{3}{2}}}} \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{k^{\frac{1}{2}} k^{\frac{3}{2}}}{k^{2} + 3} \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{k^{2}}{k^{2} + 3} \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \frac{1}{1 + \frac{3}{k^{2}}} \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 1$

Since ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{\frac{3}{2}}}$ is convergent by the p-series test so ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$ is also convergent.

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91影视

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The Limit Comparison Test.

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91影视

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.鈭�k=1鈭�kk2+3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The Limit Comparison Test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.
${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$