Q. 28 Use the alternating series test ... [FREE SOLUTION]

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Chapter 7: Q. 28 (page 653)

Use the alternating series test to determine whether the series in Exercises 24鈥�29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
${鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

The series:

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

Step 2. Rewrite the series.

By rewriting the series, we get,

${鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!} = {鈭�}_{k = 0}^{鈭�} {(- 1)}^{k + 1} \frac{k!}{(2 k + 1)!} = {鈭�}_{k = 0}^{鈭�} {(- 1)}^{k + 1} . a_{k} w h e r e a_{k} = \frac{k!}{(2 k + 1)!}$

Step 3. Find ak+1.

Consider the term,

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

So, $a_{k + 1} < a_{k}$ , so the series is monotonously decreasing.

Step 4. Determine if it converges or diverges.

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

Step 5. Calculate ak+1ak.

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{{(- 1)}^{k} (k + 1)!}{(2 k + 3)!}| = \frac{{(- 1)}^{k + 1} {(- 1)}^{2} (k + 1) 2! (2 k + 10)}{(2 k + 3) (3 k - 2)}$

Step 6. Apply the limits.

By the ratio test, the series is absolutely convergent since $r < 1$ .

So the series converges absolutely.

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91影视

Use the alternating series test to determine whether the series in Exercises 24鈥�29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
${鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1.

Step 4. Determine if it converges or diverges.

Step 5. Calculate ak+1ak.

Step 6. Apply the limits.

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91影视

Use the alternating series test to determine whether the series in Exercises 24鈥�29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.鈭�k=0鈭�(-1)k+1k!(2k+1)!

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rewrite the series.

Step 3. Find ak+1.

Step 4. Determine if it converges or diverges.

Step 5. Calculate ak+1ak.

Step 6. Apply the limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the alternating series test to determine whether the series in Exercises 24鈥�29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.
${鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 1} k!}{(2 k + 1)!}$