Q. 23 In Exercises 21-30聽use one of t... [FREE SOLUTION]

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Chapter 7: Q. 23 (page 631)

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 2}^{鈭�} \frac{3 k - 5}{k \sqrt{k^{3} - 4}}$ .

Short Answer

Expert verified

The series ${鈭�}_{k = 2}^{鈭�} \frac{3 k - 5}{k \sqrt{k^{3} - 4}}$ is convergent.

Step by step solution

Step 1. Given information

${鈭�}_{k = 2}^{鈭�} \frac{3 k - 5}{k \sqrt{k^{3} - 4}}$ .

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be the two series with positive terms then,

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ where $L$ is a positive real number then either both converge or both diverge.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ , and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ , and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges.

Step 3. The terms of the series ∑k=2∞ 3k-5kk3-4 is positive.

Find ${鈭�}_{k = 2}^{鈭�} b_{k}$ for the given series.

${鈭�}_{k = 2}^{鈭�} b_{k} = {鈭�}_{k = 2}^{鈭�} \frac{k}{k 脳 k^{\frac{3}{2}}} = {鈭�}_{k = 2}^{鈭�} \frac{1}{k^{\frac{3}{2}}}$

Step 4. Next find limk→∞ akbk for the given series.

$\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{3 k - 5}{k \sqrt{k^{3} - 4}}}{\frac{1}{k^{\frac{3}{2}}}} = \lim_{k 鈫� 鈭�} \frac{k^{\frac{3}{2}} (3 k - 5)}{k \sqrt{k^{3} - 4}} = \lim_{k 鈫� 鈭�} \frac{k^{\frac{3}{2}} 脳 k (3 - \frac{5}{k})}{k 脳 k^{\frac{3}{2}} \sqrt{1 - \frac{4}{k^{3}}}} = \lim_{k 鈫� 鈭�} \frac{(3 - \frac{5}{k})}{\sqrt{1 - \frac{4}{k^{3}}}} = 3$

Step 5. From the obtained values,

The value of $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 3$ which is a finite non zero number.

The value of ${鈭�}_{k = 2}^{鈭�} b_{k} = {鈭�}_{k = 2}^{鈭�} \frac{1}{k^{\frac{3}{2}}}$ is convergent by $p -$ series test.

Therefore, ${鈭�}_{k = 2}^{鈭�} a_{k}$ is also convergent.

Hence, the given series is convergent.

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91影视

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 2}^{鈭�} \frac{3 k - 5}{k \sqrt{k^{3} - 4}}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be the two series with positive terms then,

Step 3. The terms of the series ∑k=2∞ 3k-5kk3-4 is positive.

Step 4. Next find limk→∞ akbk for the given series.

Step 5. From the obtained values,

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91影视

In Exercises 21-30use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.鈭�k=2鈭�3k-5kk3-4.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be the two series with positive terms then,

Step 3. The terms of the series ∑k=2∞ 3k-5kk3-4 is positive.

Step 4. Next find limk→∞ akbk for the given series.

Step 5. From the obtained values,

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Pure Maths

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 2}^{鈭�} \frac{3 k - 5}{k \sqrt{k^{3} - 4}}$ .