Uniform convergence of a series on $[-1,1]$.

Question

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.

Answer 1

HINT:

$$[1+(1+x)^n]-[1+(1+x)^{n-1}]=(1+x)^{n-1}(1+x-1)$$

Reference : Telescoping series