Chapter 7: Q. 28 (page 653)

Use the alternating series test to determine whether the series in Exercises 24–29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$ converges absolutely.

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$

Step 2. Rewrite the series.

By rewriting the series, we get,

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!} = \sum_{k = 0}^{\infty} {(- 1)}^{k + 1} \frac{k!}{(2 k + 1)!} = \sum_{k = 0}^{\infty} {(- 1)}^{k + 1} . a_{k} w h e r e a_{k} = \frac{k!}{(2 k + 1)!}$

Step 3. Find ak+1.

Consider the term,

$a_{k} = \frac{k!}{(2 k + 1)!} a_{k + 1} = \frac{(k + 1)!}{(2 (k + 1) + 1)!} = \frac{(k + 1)!}{(2 k + 3)!}$

So, $a_{k + 1} < a_{k}$ , so the series is monotonously decreasing.

Step 4. Determine if it converges or diverges.

$\underset{k \to \infty}{l i m} a_{k} = a_{k} \underset{k \to \infty}{l i m} \frac{k!}{(2 k + 1)!} = a_{k} \underset{k \to \infty}{l i m} \frac{k!}{(2 k + 1) (2 k)!} = 0$

Step 5. Calculate ak+1ak.

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{{(- 1)}^{k} (k + 1)!}{(2 k + 3)!}| = \frac{{(- 1)}^{k + 1} {(- 1)}^{2} (k + 1) 2! (2 k + 10)}{(2 k + 3) (3 k - 2)}$

Step 6. Apply the limits.

By the ratio test, the series is absolutely convergent since $r < 1$ .

So the series converges absolutely.

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Use the alternating series test to determine whether the series in Exercises 24–29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1.

Step 4. Determine if it converges or diverges.

Step 5. Calculate ak+1ak.

Step 6. Apply the limits.

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Use the alternating series test to determine whether the series in Exercises 24–29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.∑k=0∞(-1)k+1k!(2k+1)!

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1.

Step 4. Determine if it converges or diverges.

Step 5. Calculate ak+1ak.

Step 6. Apply the limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Geometry

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Use the alternating series test to determine whether the series in Exercises 24–29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$