Any hints on how to simplify the following double sum to be able to find the sum at least numerically?
$$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$
Series comes from quantum mechanics (2nd order perturbation theory).