Why must Complex Fourier series be used to find Summation of $1/(n^2+1)$?

Question

Now I am not by any means skilled in this level of calculus, but I don't understand something. When I use Fourier series and calculate the coefficients by using $f(x)=e^x$, and plug in $x=\pi$, I get something completely different for $\sum_{n=1}^{\infty}\frac{1}{n^2+1}$. In fact, I obtain $\frac{\pi{e^{\pi}}}{2\sinh\pi}-\frac{1}{2}$. Can someone tell me why a different answer arises from this rather than using the complex fourier series with $c_n$? Perhaps there is something I am missing with Parseval's theorem? (P.S. I just learned this, so jargon might fly over my head. I am also new to forums completely).