Define $f(z)=\coth (z):=\frac{e^z+e^{-z}}{e^z-e^{-z}}$. The zeros of this function are $n2\pi i$ for $n\in \mathbb{Z}$, but we can holomorphically extend $\coth (z)$ to $0$. So we want to consider $f(z)$ around the origin. By 1;;15. Theorem 2.1.4 - power series expansion, we have a power series expansion for $0<r<2\pi$, and it is given by

where the $b_{2n}$ are the bernouli numbers.