It can be shown using complex analysis that,

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2+1} = \frac{\pi}{2}\left(\frac{e^{-\pi}+e^{\pi}}{e^{\pi}-e^{-\pi}}\right)-\frac{1}{2}$

Since this is a statement about real numbers, one would think that it is attainable using purely real methods, Ramanujan-style. Anyone with any ideas how to do so?