Chapter 7: Q. 32 (page 653)

Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
$\sum_{k = 1}^{\infty} \frac{{(- 5)}^{k}}{(k + 1)! - k!}$

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} \frac{{(- 5)}^{k}}{(k + 1)! - k!}$ converges absolutely.

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 1}^{\infty} \frac{{(- 5)}^{k}}{(k + 1)! - k!}$

Step 2. Rewrite the series.

$a_{k} = \frac{{(- 5)}^{k}}{(k + 1)! - k!}$

Step 3. Find ak+1

$a_{k + 1} = \frac{{(- 5)}^{k + 1}}{(k + 1 + 1)! - (k + 1)!} = \frac{{(- 5)}^{k + 1}}{(k + 2)! - (k + 1)!}$

Step 4. Calculate ak+1ak.

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{\frac{{(- 5)}^{k + 1}}{(k + 2)! - (k + 1)!}}{\frac{{(- 5)}^{k}}{(k + 1)! - k!}}| = |\frac{{(- 5)}^{k + 1} (k + 1)! - k!}{{(- 5)}^{k} (k + 2)! - (k + 1)!}| = |\frac{- 5 {(- 5)}^{k} (k + 1) k! - k!}{{(- 5)}^{k} (k + 2) (k + 1)! - (k + 1)!}| = |\frac{- 5 (k + 1 - 1) k!}{(k + 1)! ((k + 2 - 1)}| = |\frac{- 5 k . k!}{(k + 1) k! (k + 1)}| = \frac{- 5 k}{(k + 1) (k + 1)}$

Step 5. Take limits.

$\underset{k \to \infty}{l i m} |\frac{a_{k + 1}}{a_{k}}| = \underset{k \to \infty}{l i m} (\frac{- 5 k}{(k + 1) (k + 1)}) = 0$

So by the ratio test, the series converges absolutely.

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Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
$\sum_{k = 1}^{\infty} \frac{{(- 5)}^{k}}{(k + 1)! - k!}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1

Step 4. Calculate ak+1ak.

Step 5. Take limits.

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Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.∑k=1∞(-5)k(k+1)!-k!

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1

Step 4. Calculate ak+1ak.

Step 5. Take limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

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Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
$\sum_{k = 1}^{\infty} \frac{{(- 5)}^{k}}{(k + 1)! - k!}$