finding interval on which series is uniformly convergent Consider $f(x)=\sum_{n=1}^{\infty} \frac{1}{1+n^2 x}.$ 
For what values of x does it converge uniformly?
In general is there any special criterion, except Weierstrass M-test, to find out whether a Series of functions is uniformly convergent or not?

The series converges point wise for $x>0$ as well as for $x<0$ as long as $-\frac1x$ is not a perfect square (so in particular for $x<-1$).
Treating the negative part is a bit tricky, but on $(0,\infty)$, we see that $0\le f(x)\le\frac {\pi^2}{6x}$. As all summands are positive (on that interval), we have uniform convergence on any interval $[a,\infty)$ with $a>0$: Given $\epsilon>0$, we certainly have an error $<\epsilon$ for $x>\frac{\pi^2}{6\epsilon}$, hence need only concentrate on the interval $[a,\frac{\pi^2}{6\epsilon}]$, which is compact. 
In fact, we only used "positive continuous summands with a limit tending to $0$ at infinity". The same trick can be used to show that $f$, with the finitely many summands having $1+n^2a\ge 0$ removed, converges uniformly on $(-\infty,a]$ for any $a<0$. Thus after adding bac those finitely many summands, $f$ converges uniformly on any closed subinterval of $\Bbb R\setminus A$ where $A=\{0\}\cup\{\,-\frac1{n^2}\mid n\in\Bbb N\,\}$ is the set of points where $f$ does not converge (or even some summand is undefined) and tends to infinity.
