Q. 31 In Exercises 29鈥�34 use the rat... [FREE SOLUTION]

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Chapter 7: Q. 31 (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.
${鈭�}_{k = 0}^{鈭�} \frac{1}{k^{3} + 1}$

Short Answer

Expert verified

The series converges.

Step by step solution

Step 1. Given information.

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

Step 2. Ratio Test.

Now,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{1}{(k + 1)^{3} + 1}}{\frac{1}{k^{3} + 1}} = \frac{k^{3} + 1}{(k + 1)^{3} + 1} = \frac{k^{3} + 1}{k^{3} + 1 + 3 k^{2} + 3 k + 1} \frac{a_{k + 1}}{a_{k}} = \frac{k^{3} + 1}{k^{3} + 3 k^{2} + 3 k + 2} \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = \lim_{k 鈫� 鈭�} \frac{k^{3} + 1}{k^{3} + 3 k^{2} + 3 k + 2} = (\lim_{k 鈫� 鈭�} \frac{k^{3} (1 + \frac{1}{k^{3}})}{k^{3} (1 + \frac{3}{k} + \frac{3}{k^{2}} + \frac{2}{k^{3}})}) = 1 T e s t I n c o n c l u s i v e .$

Step 3. Conclusion.

Now

${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k^{3}} is of the form {鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k^{p}} which is a p -series. Here, p = 3 > 1 .$

Hence, the series converges.

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Conclusion.

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91影视

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=0鈭�1k3+1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Geometry

Decision Maths

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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