<s>[INST] Uniform convergence of $\sum_{n=1}^{\infty}\frac{x^n}{1+x^n}$ 
Where does the seriers $\displaystyle \sum_{n=1}^{\infty}\frac{x^n}{1+x^n}$ converges ? On which interval does it converge uniformly ?

We have $$\lim_n \frac{x^n}{1+x^n}=\begin{cases}0 &\text{ , if } |x|<1\\\frac{1}{2} &\text{ , if } x=1\\\text{does not exists} &\text{ , if } x=-1\\1 &\text{ , if } |x|>1\end{cases}$$
So we have to test the convergence only on the interval $(-1,1)$.
Now let , $\displaystyle S_n=\sum_{k=1}^n \frac{x^k}{1+x^k}$. How I find the point wise sum ? 
Edit: I'm confused for different answers and different comments..!!
 [/INST] Observe for all $|x| \leq R<1$ you have that
\begin{align}
\left|\sum^m_{n=k} \frac{x^n}{1+x^n}\right| \leq \sum^m_{n=k} \frac{|x|^n}{1-R^n} \leq \frac{1}{1-R}\sum^m_{n=k}R^n = \frac{R^k-R^{m+k+1}}{(1-R)^2}
\end{align}
which means  the series is uniformly Cauchy on $[-R, R]$. Hence the series is uniformly convergent on any compact subset of $(-1, 1)$. 
Edit: Suppose the series is uniformly convergent on all of $(-1, 1)$. Fix $\epsilon = \frac{1}{2}$, then there exists $N$ such that
\begin{align}
\left|\sum^\infty_{n=N}\frac{x^n}{1+x^n}\right| <\frac{1}{2}
\end{align}
for all $x \in (-1, 1)$. Let $x_\ell = \frac{\ell-1}{\ell}$, then observe that
\begin{align}
\sum^\infty_{n=N} \frac{x_\ell^n}{1+x_\ell^n} = \sum^\infty_{n=N} \frac{(\ell-1)^n}{\ell^n+(\ell-1)^n} \geq \frac{1}{2}\sum^\infty_{n=N}\left(\frac{\ell-1}{\ell}\right)^n = \frac{1}{2}\frac{\left(\frac{\ell-1}{\ell}\right)^N}{1-\frac{\ell-1}{\ell}} = \frac{\ell}{2}\left(1-\frac{1}{\ell} \right)^N \rightarrow \infty
\end{align}
as $\ell\rightarrow \infty$. 
</s>