Chapter 7: Q. 35 (page 653)

Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$ converges absolutely.

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$

Step 2. Rewrite the series.

$a_{k} = \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$

Step 3. Find ak+1.

$a_{k + 1} = \frac{{(- 1)}^{k + 1} 1.3.5 . . . (2 (k + 1) + 1)}{1.4.7 . . . (3 (k + 1) + 1)} = \frac{{(- 1)}^{k + 1} 1.3.5 . . . (2 k + 3)}{1.4.7 . . . ((3 k + 4)}$

Step 4. Calculate ak+1ak.

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{\frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 3)}{1.4.7 . . . ((3 k + 4)}}{\frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}}| = |\frac{(- 1) 2 k + 3}{3 k + 4}| = \frac{2 k + 3}{3 k + 4}$

Step 5. Take limits.

$\underset{k \to \infty}{l i m} |\frac{a_{k + 1}}{a_{k}}| = \underset{k \to \infty}{l i m} (\frac{(2 k + 3)}{3 k + 4}) = \underset{k \to \infty}{l i m} (\frac{k (2 + \frac{3}{k})}{k (3 + \frac{4}{k})}) = (\frac{2}{3}) = \frac{2}{3} < 1$

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.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$

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∞(-1)k1.3.5...(2k+1)1.4.7...((3k+1)

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

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

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Applied Mathematics

Logic and Functions

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.
$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} 1.3.5 . . . (2 k + 1)}{1.4.7 . . . ((3 k + 1)}$