Basel sum from exponential poles (${\sum }_{r\ge 1}{r}^{-2}={\pi }^2/6$)

The meromorphic function $(e^z-1{)}^{-1}$ has simple poles of residue one at $2\pi i\mathbb{Z}$. Subtracting $1/z+{\sum }_{r=1}^{n}2z/(z^2+4{\pi }^2{r}^2)$ removes all poles inside radius $2(n+1)\pi$. Comparing its inner Taylor coefficients with the outer Laurent expansion gives annular coefficient $a_1=1/12-{\sum }_{r=1}^{n}1/(2{\pi }^2{r}^2)$. On circles of radius $(2n+1)\pi$, the function is uniformly bounded: outside the strip $|\mathrm{Re}z|\le 1$ use the exponential modulus, and inside it the imaginary part stays near an odd multiple of $\pi$. The Laurent coefficient integral bounds $|a_1|\le C/[(2n+1)\pi ]\to 0$, proving the sum.