How to prove that $\sum\limits_{n=1}^\infty \frac{x}{x^2+n^2}$ is uniformly convergent for every $x$?
I was trying all sort of ways, but it think the answer might be in solving the problem for $|x|<1$ and then for $|x|>1$.
Its easy to show that for $|x|<1$ , $\sum\limits_{n=1}^\infty \frac{x}{x^2+n^2} \leq \sum\limits_{n=1}^\infty \frac{1}{n^2}$ and using Weierstrass M-Test that the sum is uniformly convergent.
But for $|x|>1$ its a different story.
Does any one have a simple solution? I'm stuck...