Chapter 7: Q. 51 (page 625)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information is:

$\sum_{k = 1}^{\infty} \frac{k}{{(1 + k^{2})}^{p}}$

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:

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

Step 4. Result:

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

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

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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In Exercises 48–51 find all values of p so that the series converges.∑k=1∞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

Theoretical and Mathematical Physics

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

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