Chapter 7: Q. 32 (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
$\sum_{k = 1}^{\infty} \frac{k!}{(2 k)!}$

Short Answer

Expert verified

The series converges.

Step by step solution

Step 1. Given information.

The given series is $\sum_{k = 1}^{\infty} \frac{k!}{(2 k)!} .$

Step 2. Root test.

According to the series,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{(k + 1)!}{(2 k + 2)!}}{\frac{k!}{(2 k)!}} = \frac{(2 k)! (k + 1)!}{k! (2 k + 2)!} \frac{a_{k + 1}}{a_{k}} = \frac{(2 k)! (k + 1) k!}{k! (2 k + 2) (2 k + 1) (2 k)!} = \frac{(k + 1)}{2 (k + 1) (2 k + 1)} = \frac{1}{2 (2 k + 1)}$

Step 3. Conclusion.

On taking limits,

$\lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} \frac{1}{2 (2 k + 1)} = \frac{1}{2} (\lim_{k \to \infty} \frac{1}{2 k + 1}) = 0 Since, L < 1,$

Therefore, 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
$\sum_{k = 1}^{\infty} \frac{k!}{(2 k)!}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Root 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=1∞k!(2k)!

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Root test.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Pure Maths

Decision Maths

Applied Mathematics

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
$\sum_{k = 1}^{\infty} \frac{k!}{(2 k)!}$