TITLE: Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$?
QUESTION [5 upvotes]: Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$?
I have no idea where to start. Could somebody give me  a hint ?
Edit: Could I use something like this ?
$$\left|\frac{x^n}{1+x^n}\right|\leq x^n$$
Because $x<1$, the geometric series $\sum x^n$ converges. Therefore by the M-test we get that the series converges uniformly.

REPLY [4 votes]: By definition the serie $\displaystyle\sum\frac{x^n}{1+x^n}$ converges uniformly on $[0,1)$ if 
$$\lim_n\sup_{x\in[0,1)} \sum_{k=n+1}^{\infty}\frac{x^k}{1+x^k}=0.$$
We have 
$$\sum_{k=n+1}^{\infty}\frac{x^k}{1+x^k}\geq \sum_{k=n+1}^{\infty}\frac{x^k}{2}=\frac{1}{2}\frac{x^{n+1}}{1-x},$$
So it's clear that $\displaystyle\sup_{x\in[0,1)} \frac{x^{n+1}}{1-x}=+\infty, $ and hence the condition of uniform convergence is not verified.