Chapter 9 - Infinite Series - 9.3 Exercises - Page 609: 13

By theorem 9.3, since the integral does not equal infinity, the series converges.

$\Sigma^\infty_{n=1} \frac{arctan(x)}{n^{2}+1}$ $Let f(x) = \frac{arctan(x)}{n^{2}+1}$ $f’(x) = \frac{1-2xarctan(x)}{({x^{2}+1})^2} \lt $ 0 for all x $\geq 1$ f is positive, continuous and decreasing for x $\geq$ 1 $\int^\infty_1 \frac{arctan(x)}{x^{2}+1} dx = [\frac{{(arctan(x))}^{2}}{2}]^\infty_1 = \frac{3\pi^{2}}{32}$