pointwise and uniformly convergence of a series of functions $f_n(x)$ I have $\sum \frac{e^{nx}}{2+e^{nx}} arctan(n^{|x|})$. I have pointwise convergence in $(-\infty,0)$.But for the uniform convergence? 
If I consider $\sum sup_{(-\infty,0)} |f_n(x)|\le \sum \frac{\pi}{2} \frac{e^{nx}}{2+e^{nx}}=\frac {\pi}{4}$ series not convergent

HINT:
For any fixed $N$ with $x=1/N$, we have that 
$$\begin{align}
\left|\sum_{n=0}^\infty \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)-\sum_{n=0}^{N} \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)\right|&=\left|\sum_{n=N+1}^\infty \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)\right|\\\\
&=\sum_{n=N+1}^\infty \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)\\\\
&=\sum_{n=N+1}^\infty \frac{e^{-n/N}}{2+e^{-n/N}}\arctan\left(n^{1/N}\right)\\\\
&\ge \frac{\pi}{12}\sum_{n=N+1}^\infty e^{-n/N}\\\\
&\ge \frac{\pi }{12e(e-1)}
\end{align}$$
