Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$ How can I prove that for every $ x \notin \mathbb Z$ the series
$$ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$$
converges uniformly in a neighborhood of $ x $?

Let $k\in\mathbb{Z}$ and $a,b\in\mathbb{R}$ such that $k<a<b<k+1$ then we have:
$$\forall x\in[a,b],\quad\frac{1}{|n^2-x^2|}\leq \max\left(\frac{1}{|n^2-a^2|},\frac{1}{|n^2-b^2|}\right)\sim_\infty\frac{1}{n^2}$$
so the series is normal convergent on the interval $[a,b]$ and hence it's uniform convergent on $[a,b]$.
