Chapter 7: Q. 30 (page 653)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

The series:

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

Step 2. Rewrite the series.

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

Step 3. Find ak+1.

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

Step 4. Calculate ak+1ak.

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

Step 5. Take limits.

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

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

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∞(-k)kk!

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

Decision Maths

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Statistics

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Calculus

Theoretical and Mathematical Physics

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