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Chapter 7: Q. 34 (page 639)

In Exercises 29鈥�34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.
${鈭�}_{k = 1}^{鈭�} \frac{1 路 4 路 7 鈰� (3 k - 2)}{2 路 4 路 6 鈰� (2 k)}$

Short Answer

Expert verified

The series diverges.

Step by step solution

01

Step 1. Given information.

The given series is ${鈭�}_{k = 1}^{鈭�} \frac{1 路 4 路 7 鈰� (3 k - 2)}{2 路 4 路 6 鈰� (2 k)} .$

02

Step 2. Ratio Test.

Now,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{1 路 4 路 7 路鈰� (3 k - 2) (3 k + 1)}{2 路 4 路 6 路鈥� (2 k) (2 k + 2)}}{\frac{1 路 4 路 7 鈰� (3 k - 2)}{2 路 4 路 6 鈥� (2 k)}} = \frac{3 k + 1}{2 k} \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = \lim_{k 鈫� 鈭�} \frac{3 k + 1}{2 k} = \lim_{k 鈫� 鈭�} \frac{k (3 + \frac{1}{k})}{k (2)} = \frac{3}{2} Since, L > 1$

Therefore, the series diverges.

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Most popular questions from this chapter

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges to L and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges to M , then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

37. ${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$

Determine whether the series ${鈭�}_{j = 2}^{鈭�} \frac{2^{j}}{3}$ converges or diverges. Give the sum of the convergent series.

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} \frac{1}{k}$

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