Integral Test — Question 4

*Question 4 Consider $$\sum_{n=1}^{\infty}\frac{n}{(n^2+1)^2}.$$

1. For $f(x)=x/(x^2+1)^2$, verify positivity, continuity, and decreasing behavior on $[1,\infty)$.

2. Evaluate the associated improper integral using an appropriate substitution.

3. Determine convergence and obtain upper and lower bounds for $R_N$.

Clearly distinguish the value of the integral from the value of the series.

*Question 4 – Solution

Let $f(x)=x/(x^2+1)^2$ for $x\ge1$. It is continuous and positive. Differentiation gives $$f'(x)=\frac{1-3x^2}{(x^2+1)^3}<0\qquad(x\ge1),$$ so it decreases.

With $u=x^2+1$ and $du=2x\,dx$, $$\int_1^{\infty}\frac{x}{(x^2+1)^2}\,dx =\frac12\int_2^{\infty}u^{-2}\,du=\frac14.$$ The integral converges, so the series converges.

Since $$\int_A^{\infty}\frac{x}{(x^2+1)^2}\,dx=\frac1{2(A^2+1)},$$ the remainder satisfies $$\boxed{\frac1{2((N+1)^2+1)}\le R_N\le\frac1{2(N^2+1)}}.$$