continuity of a function series I need help finding for which $x$ the function:
$$\sum_{n=1}^{\infty}\frac{x+n(-1)^n}{x^2+n^2}$$
is continuous. 
I first need to show that the series converges uniformly, the thing is I don't know how to deal with the $(-1)^n$. I tried comparing to the series $\sum_{n=1}^{\infty}\frac{x+n}{x^2+n^2}$ but I think I hit a dead end there.
any suggestions? 

as I mentioned in Doug answer, if we will prove the series coverges uniformly we can easily show it's contionous. 
when $n\rightarrow \infty$ than the series acts as $\sum_{n=1}^{\infty}(-1)^n\frac{n}{x^2+n^2}$. using Leibniz test we know the series converges so from that we conclude that for all x's the series is continuous.
