Chapter 7: Q. 31 (page 653)

Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
$\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{(2 k)!}$

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{(2 k)!}$ converges absolutely.

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{(2 k)!}$

Step 2. Rewrite the series.

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

Step 3. Find ak+1.

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

Step 4. Calculate ak+1ak.

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

Step 5. Take limits.

$\underset{k \to \infty}{l i m} |\frac{a_{k + 1}}{a_{k}}| = \underset{k \to \infty}{l i m} (\frac{3}{2 (k + 1) (2 k + 1)}) = 0$

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.
$\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{(2 k)!}$

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=0∞(-3)k+1(2k)!

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

Mechanics Maths

Pure Maths

Applied Mathematics

Statistics

Geometry

Probability and Statistics

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.
$\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{(2 k)!}$