Chapter 7: Q. 24 (page 652)

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

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} \frac{(- k)^{k}}{k!}$ diverges

Step by step solution

Step 1. Given information

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

Step 2. Substitute k=k+1 inak=(k)kk!

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

Now,

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

Step 3. Adding the limit

$\begin{aligned} lim_{k \to \infty} a_{k} = lim_{k \to \infty} \frac{(k)^{k}}{k!} \\ \neq 0 \end{aligned}$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Substitute k=k+1 inak=(k)kk!

Step 3. Adding the limit

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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=1∞ (−k)kk!

Short Answer

Step by step solution

Step 1. Given information

Step 2. Substitute k=k+1 inak=(k)kk!

Step 3. Adding the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Probability and Statistics

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