Compute $\sum_{n\geq1}\frac{1}{\pi^2+n^2}$ by expanding $e^{\pi x}$ in its Fourier series

Question

Compute $$\sum_{n\geq1}\frac{1}{\pi^2+n^2}.$$

by expanding $e^{\pi x}$ in its Fourier series.

So I calculated that

$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{\pi x} e^{-inx} \ dx=\frac{\sinh{(\pi^2)}(-1)^n}{\pi(\pi-in)},$$

which is correct. Now,

$$f(x)=e^{\pi x}=\sum_{n\in{\mathbb{Z}}}\frac{\sinh{(\pi^2)}(-1)^n}{\pi(\pi-in)}e^{inx}.$$

At $x=0$ we have

$$1=\sum_{n\in{\mathbb{Z}}}\frac{\sinh{(\pi^2)}(-1)^n}{\pi(\pi-in)}=\frac{\sinh(\pi^2)}{\pi}\sum_{n\in\mathbb{Z}}\frac{(-1)^n}{\pi-in}.$$

Taking out the case $n=0$ and splitting the sum by $n \rightarrow -n$ in one of them we get

\begin{align} 1&=\frac{\sinh(\pi^2)}{\pi}\left(\frac{1}{\pi}+\sum_{n\geq1}\frac{(-1)^n}{\pi-in} + \sum_{n\geq1}\frac{(-1)^{-n}}{\pi+in}\right)\\ &=\frac{\sinh(\pi^2)}{\pi}\left(\frac{1}{\pi}+2\pi\sum_{n\geq1}\frac{(-1)^n}{\pi^2+n^2}\right). \end{align}

So I almost have the desired sum here, only thing bothering me is the $(-1)^n$. How do I proceed?

Note that I'm not interested in alternate solutions. I already know at least 2 other methods.

Answer 1

Just to put in into an answer rather than into a comment I have to say that I am almost sure the only, and most likely intended way is to evaluate the series at $x=\pi$, as pointed out by Semiclassical. This is yet alone the only possibilty to get rid of the $(-1)^n$ since this factor has nothing to do with the original function but more with underlying structure of the Fourier Coefficients.

A typical approach to find oscillating series, such as the one you derived from the non-oscillating one, is by playing the good old even odd cancelation game $($here similiar is done with the Riemann Zeta Function and its relative, the Dirichlet Eta Function$)$. However, applying this method to the given problem, and ignoring the constant factors for a moment, we get the following

$$\underbrace{\sum_{n\ge1}\frac{(-1)^n}{\pi^2+n^2}}_{=S_1}=\underbrace{\sum_{n\ge1}\frac1{\pi^2+n^2}}_{=S}-\underbrace{2\sum_{n\ge1}\frac1{\pi^2+(2n)^2}}_{=S_2}$$

Our main goal is to find $S$ wherease $S_1$ can be computed by your given formula. What remains to find is an expression for $S_2$. And from hereon we cannot get any further by only taking the Fourier Series Expansion of $f(x)=e^{\pi x}$, at least as far as I can tell. One could use the expansion of $F(x)=e^{ax}$, for a positive real number $a$, instead and it is quite funny following this ansatz since it is way off what I think was intended by this question. Computing $c_n$ for $F(x)$ we obtain

$$F(x)=\frac{\sinh(a\pi)}\pi\sum_{n\in\mathbb Z}\frac{\color{red}{(-1)^n}}{a-in}e^{-inx}$$

Well, there is it again, the oscillating minus sign. However, note that for $a=\frac\pi2$ we are actually able to deduce an expression for $S_2$; by evaluating at $x=\pi$ ... Moreover I am certainly sure that this is not the expected method, even though it works afterall.

Long story short: to deal with oscillating minus signs is easily done by splitting the sum up into even and odd parts but sometimes it makes the problem even more complicated than it actually is.