Chapter 7: Q. 31 (page 639)

In Exercises 29–34 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.
$\sum_{k = 0}^{\infty} \frac{1}{k^{3} + 1}$

Short Answer

Expert verified

The series converges.

Step by step solution

Step 1. Given information.

The given series is $\sum_{k = 0}^{\infty} \frac{1}{k^{3} + 1} .$

Step 2. Ratio Test.

Now,

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

Step 3. Conclusion.

Now

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

Hence, the series converges.

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In Exercises 29–34 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.
$\sum_{k = 0}^{\infty} \frac{1}{k^{3} + 1}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Conclusion.

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In Exercises 29–34 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.∑k=0∞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

Calculus

Discrete Mathematics

Geometry

Mechanics Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercises 29–34 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.
$\sum_{k = 0}^{\infty} \frac{1}{k^{3} + 1}$