Q. 10 Explain why the ratio test does ... [FREE SOLUTION]

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Chapter 7: Q. 10 (page 639)

Explain why the ratio test does not work in determining the convergence or divergence of the series ${鈭�}_{k = 1}^{鈭�} \frac{k!}{(k + 2)!}$ . What test would be more effective to analyze this series?

Short Answer

Expert verified

Hence proved that the ratio test will be inconclusive because $L = 1$ and Convergence and Divergence test can be used.

Step by step solution

Step 1. Given information.

We are given ${鈭�}_{k = 1}^{鈭�} \frac{k!}{(k + 2)!}$ .

Step 2. Ratio Test.

On using Ratio Test,

$a_{k + 1} = \frac{(k + 1)!}{(k + 3)!} \frac{a_{k + 1}}{a_{k}} = \frac{\frac{(k + 1)!}{(k + 3)!}}{\frac{k!}{(k + 2)!}} \frac{a_{k + 1}}{a_{k}} = \frac{(k + 1) k! (k + 2)!}{(k + 3) (k + 2)! k!} = \frac{k + 1}{k + 3} \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = \lim_{k 鈫� 鈭�} \frac{k + 1}{k + 3} = \lim_{k 鈫� 鈭�} \frac{k (1 + \frac{1}{k})}{k (1 + \frac{3}{k})} L = 1 T h e r e f o r e, t h e t e s t i s i n c o n c l u s i v e .$

Step 3. Convergence and Divergence test.

Now,

${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{k!}{(k + 2)!} and {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} Clearly, 0 鈮� \frac{k!}{(k + 2)!} 鈮� \frac{1}{k^{2}} . Now, {鈭�}_{k = 1}^{鈭�} b_{k} - {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} is of the form {鈭�}_{k = 1}^{鈭�} b_{k} - {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{p}} which ie a p eeries. Here, p = 2 > 1 . Hence, the series {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} converges.$

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91影视

Explain why the ratio test does not work in determining the convergence or divergence of the series ${鈭�}_{k = 1}^{鈭�} \frac{k!}{(k + 2)!}$ . What test would be more effective to analyze this series?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Convergence and Divergence test.

One App. One Place for Learning.

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91影视

Explain why the ratio test does not work in determining the convergence or divergence of the series 鈭�k=1鈭�k!(k+2)!. What test would be more effective to analyze this series?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Convergence and Divergence test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Explain why the ratio test does not work in determining the convergence or divergence of the series ${鈭�}_{k = 1}^{鈭�} \frac{k!}{(k + 2)!}$ . What test would be more effective to analyze this series?