Q. 51 In Exercises 48鈥�51 find all va... [FREE SOLUTION]

Chapter 7: Q. 51 (page 625)

In Exercises 48鈥�51 find all values of p so that the series converges.
${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Short Answer

Expert verified

$The integral {鈭�}_{x = 1}^{鈭�} \frac{x}{{(1 + x^{2})}^{p}} dx converges for p > 1 . Thus, the series {鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}} is convergent for p > 1 .$

Step by step solution

Step 1. Given information is:

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

Step 2. Examining nature of given function:

$Consider the function f (x) = \frac{x}{{(1 + x^{2})}^{p}} . The function f (x) = \frac{x}{{(1 + x^{2})}^{p}} is continuous, decreasing, with positive terms . Therefore all the conditions of integral test are fulfilled . So, integral test is applicable .$

Step 3. Solving the integral:

$Consider the integral : {鈭�}_{x = 1}^{鈭�} f (x) dx = {鈭�}_{x = 1}^{鈭�} \frac{x}{{(1 + x^{2})}^{p}} dx . Therefore, {鈭�}_{x = 1}^{鈭�} f (x) dx = \lim_{k 鈫� 鈭�} {鈭�}_{x = 1}^{k} \frac{x}{{(1 + x^{2})}^{p}} dx = \frac{1}{2} \lim_{k 鈫� 鈭�} {鈭�}_{u = 2}^{k^{2} + 1} \frac{du}{u^{p}} (Put 1 + x^{2} = u, 鈬� 2 xdx = du) = \frac{1}{2} \lim_{k 鈫� 鈭�} {[\frac{u^{- p + 1}}{- p + 1}]}_{2}^{k^{2} + 1} (Integrating) = \frac{1}{2 (- p + 1)} \lim_{k 鈫� 鈭�} {[\frac{1}{u^{p - 1}}]}_{2}^{k^{2} + 1} (Substitution)$

Step 4. Result:

$The improper integral converges to finite value only when p > 1 . Therefore, the integral {鈭�}_{x = 1}^{鈭�} \frac{x}{{(1 + x^{2})}^{p}} dx converges for p > 1 . Thus, the series {鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}} is convergent for p > 1 .$

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91影视

In Exercises 48鈥�51 find all values of p so that the series converges.
${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Examining nature of given function:

Step 3. Solving the integral:

Step 4. Result:

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91影视

In Exercises 48鈥�51 find all values of p so that the series converges.鈭�k=1鈭�k(1+k2)p

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Examining nature of given function:

Step 3. Solving the integral:

Step 4. Result:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 48鈥�51 find all values of p so that the series converges.
${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$