Uniform convergence of a series on $[-1,1]$. I am stuck with this problem. I  am trying to check the uniform convergence of the series
$$\sum_{n=1}^{\infty} \dfrac{-2x(1+x)^{n-1}}{[1+(1+x)^{n-1}][1+(1+x)^n]}$$
on the interval $[-1,1]$ and $[0,1]$.
The first thing I could think was to of course put $t= 1+x$. I thought resolving into partial fractions would be the next step. But that is not proving to be useful the way I have done. Am I doing this correct or should I adopt another approach. 
P.S. This is not a homework problem. I am just trying some problems on my own.

HINT:
$$[1+(1+x)^n]-[1+(1+x)^{n-1}]=(1+x)^{n-1}(1+x-1)$$
Reference : Telescoping series
