Q. 21 In Exercises 21鈥�30 use one of ... [FREE SOLUTION]

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Chapter 7: Q. 21 (page 631)

In Exercises 21鈥�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.
${鈭�}_{k = 0}^{鈭�} \frac{3 k^{2} + 1}{k^{3} + k^{2} + 5}$ .

Short Answer

Expert verified

The series ${鈭�}_{k = 0}^{鈭�} \frac{3 k^{2} + 1}{k^{3} + k^{2} + 5}$ is divergent.

Step by step solution

Step 1. Given information

${鈭�}_{k = 0}^{鈭�} \frac{3 k^{2} + 1}{k^{3} + k^{2} + 5}$ .

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

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where $L$ is any 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 ${鈭� a_{k}}_{k = 1}^{鈭�}$ converges.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges, then ${鈭� a_{k}}_{k = 1}^{鈭�}$ diverges.

Step 3. The term series of the ∑k=0∞3k2+1k3+k2+5 is positive.

Find ${鈭� b_{k}}_{k = 0}^{鈭�}$ for the given series.

${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{k^{2}}{k^{3}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k}$

Step 4. Next find limk→∞akbkfor the given series.

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

Step 5. From the obtained values,

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

The value of ${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k}$ is divergent by $p -$ series test.

Therefore, the series ${鈭�}_{k = 0}^{鈭�} a_{k}$ is also divergent.

Hence the given series is divergent.

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91影视

In Exercises 21鈥�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.
${鈭�}_{k = 0}^{鈭�} \frac{3 k^{2} + 1}{k^{3} + k^{2} + 5}$ .

Short Answer

Step by step solution

Step 1. Given information

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

Step 3. The term series of the ∑k=0∞3k2+1k3+k2+5 is positive.

Step 4. Next find limk→∞akbkfor the given series.

Step 5. From the obtained values,

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91影视

In Exercises 21鈥�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=0鈭�3k2+1k3+k2+5.

Short Answer

Step by step solution

Step 1. Given information

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

Step 3. The term series of the ∑k=0∞3k2+1k3+k2+5 is positive.

Step 4. Next find limk→∞akbkfor the given series.

Step 5. From the obtained values,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Discrete Mathematics

Statistics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 21鈥�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.
${鈭�}_{k = 0}^{鈭�} \frac{3 k^{2} + 1}{k^{3} + k^{2} + 5}$ .