Checking where a series of functions is uniformly convergent, an application of Weierstrass M-Test?

Question

Let $\sum_{n=1}^{\infty}$fn(x) = $\sum_{n=1}^{\infty} \frac{(1-x^2)^n}{(1+x^2)^n}sin(nx)$
I am attempting to figure out where this series is uniformly and absolutely convergent.
I am trying to use M-Test,
if 0 < x < 1 then $0<x^2<1$
so |$\frac{(1-x^2)^n}{(1+x^2)^n}sin(nx)|<\frac{1}{(1+x^2)^n} $
the RHS is convergent geometric series, so I applied M-test.
Is this the correct approach to use?
And how do I check that this series is not uniformly convergent on x>1.

Answer 1

Partial answer. The series is not uniformly convergent on $(1,\infty)$ and $(-\infty, -1)$: If it is uniformly convergent there then the general term must tend to $0$ uniformly. But then $\sin (nx) \to 0$ uniformly. Take $x=\frac {2m\pi} n+\frac {\pi} {2n}$ with $m$ large to get a contradiction.

By M-test the series converges uniformly on $\{x: \epsilon \leq |x| \leq 1\}$ for any $\epsilon \in (0,1)$.