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Chapter 7: Q. 33 (page 653)

Use the ratio test for absolute convergence to determine whether the series in Exercises 30鈥�35 converge absolutely or diverge.
${鈭�}_{k = 0}^{鈭�} \frac{{(- 7)}^{k}}{(2 k + 1)!}$

Short Answer

Expert verified

The series ${鈭�}_{k = 0}^{鈭�} \frac{{(- 7)}^{k}}{(2 k + 1)!}$ converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

${鈭�}_{k = 0}^{鈭�} \frac{{(- 7)}^{k}}{(2 k + 1)!}$

02

Step 2. Rewrite the series.

$a_{k} = \frac{{(- 7)}^{k}}{(2 k + 1)!}$

03

Step 3. Find ak+1.

$a_{k + 1} = \frac{{(- 7)}^{k + 1}}{(2 (k + 1) + 1)!} = \frac{{(- 7)}^{k + 1}}{(2 k + 2 + 1)!} = \frac{{(- 7)}^{k + 1}}{(2 k + 3)!}$

04

Step 4. Calculate ak+1ak.

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{\frac{{(- 7)}^{k + 1}}{(2 k + 3)!}}{\frac{{(- 7)}^{k}}{(2 k + 1)!}}| = |\frac{{(- 7)}^{k + 1} (2 k + 1)!}{{(- 7)}^{k} (2 k + 3)!}| = |\frac{- 7 {(- 7)}^{k} (2 k + 1)!}{{(- 7)}^{k} (2 k + 3) (2 k + 2) (2 k + 1)!}| = |\frac{- 7}{(2 k + 3) (2 k + 2)}| = \frac{7}{2 (2 k + 3) (k + 1)}$

05

Step 5. Take limits.

$\underset{k 鈫� 鈭�}{l i m} |\frac{a_{k + 1}}{a_{k}}| = \underset{k 鈫� 鈭�}{l i m} \frac{7}{2 (2 k + 3) (k + 1)} = 0$

So by the ratio test, the series converges absolutely.

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Most popular questions from this chapter

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{\tan^{鈭� 1} 鈦� k}{1 + k^{2}}$

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Given a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ , in general the divergence test is inconclusive when . For a $a_{k} 鈫� 0$ geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

What is the contrapositive of the implication 鈥淚f A, then B"?

Find the contrapositives of the following implications:

If a divides b and b dividesc, then a divides c.

See all solutions

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