Boundary Values & Fourier Series · Challenge problem

The Fourier series of $e^x$ on $[-\pi,\pi]$ has $a_n = \dfrac{2(-1)^n\sinh\pi}{\pi(1+n^2)}$ for $n \ge 0$. By evaluating the series at a suitable point, find $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{1+n^2}$. Give the value as a decimal rounded to $4$ places.