There are many proofs out there for $\sum\_{n\in N} n^{-2}=\frac{\pi^2}{6}$. I am looking for a proof/reference of the following way that uses convergence Fourier series. First using the Poisson kernel on $R$, given by $k(x)=\frac{1}{\pi(1+x^2)}$ we get derive an identity
$$\sum\_{n\in Z}(n^2+t^2)^{-1}=\frac{\pi}{t}\frac{1+e^{-2\pi t}}{1-e^{-2\pi t}}$$
and then taking limits of the above as $t\to 0$ we deduce the result. However I dont understand where this identity comes from and the limit part, what technique is used.