Pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ $x\in R$

Question

How can I prove pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ for $x\in \mathbb{R}$?

uniform convergence: the pointwise convergence is on $E=(-\infty,0]$.

$f_n(x)={\frac{x}{x+e^{-nx}}}$

sup$_{E}|{\frac{x}{x+e^{-nx}}}|=f_n(\frac{-1}{n})={\frac{1}{1-ne}}$ general term of a divergent series.So there is not uniform convergent in E. But if I consider intervals as $(-\infty,a],a<0$ there is uniform convergence?

Answer 1

Series is not even point wise convergent . Look at $x=1$.

Answer 2

You can only try to use Leibnitz criterion if $x<0$ because otherwise $|a_n|$ is not decreasing and convergent to zero. In any case, for $x >0$, the series is divergent as the general term does not go to zero.

Answer 3

For $x>0$, the general term is asymptotic to $(-1)^n$ (the exponential vanishes) and the series diverges.

For $x=0$ the general term is $0$.

For $x<0$, the exponential dominates and the general term is asymptotic to $xe^{nx}$, and the series converges.