Question

Use the integral test to find whether the following series converge or diverge. Co (n2 + 1)2

Math Physics:

IMPORTANT! Step by step *typed* format preferred. If hand written, please write clearly and legibly. Please be sure answer is CORRECT! (No wrong answers please!) Thanks!

Answer 1

The series will converge if the following integral will converge,

The series will converge if the following integral will converge, integral^infinity_1 x/(x^2 + 1)^2 dx Evaluating the integral, We will take, t = x^2 + 1 = > dt = 2 x dx The limits will change to 2 to infinity. Substituting we get, integral^infinity_2 dt/2t^2 = 1/2 integral^infinity_2 dt/t^2 = -1/2 Vertical-bar 1/t Vertical-bar ^infinity_2 = -1/2 (1/infinity - 1/2) = 1/4 Since the integral converges to a value, therefore the above series converges to a value.

Evaluating the integral,

We will take,

$t = x^2 + 1 \Rightarrow dt = 2xdx$

The limits will change to 2 to infinity.

Substituting we get,

$\int_2^\infty \frac{dt}{2t^2} = \frac{1}{2}\int_2^\infty \frac{dt}{t^2} = -\frac{1}{2} \left |\frac{1}{t} \right |^\infty _2 = -\frac{1}{2} \left(\frac{1}{\infty} - \frac{1}{2} \right ) = \frac{1}{4}$

Since the integral converges to a value, therefore the above series converges to a value.