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?

Suppose $x\geq 0$ to avoid values like $\frac{-1}{4}, \frac{-1}{9},...$.
For $a>0$, the series converges uniformly at $[a,+\infty)$ by M-test.
it doesn't at $[0,+\infty)$ cause if the concergence is uniform at $[0,+\infty)$, the sum function will be continuous at $x=0$ cause the functions $x\mapsto \frac{1}{1+n^2x}$ are all continuous at $x=0$. but for $x=0$, the series diverges.
