Trouble with proving uniform convergence prove : $$\sum_{n=1}^{\infty } \dfrac{nx}{e^{nx}}$$
converge for each x $\in R$ , converge uniformly in $[\alpha,\infty)$ for $\alpha >0$ and doesn't converger uniformly at $[0,\infty)$. 
Having trouble with that question, 
Thanks for helping.

Note that $z/e^z$ has a maximum value at $z=1$ and is decreasing for $z > 1$. 
We have for all $n > 1/\alpha$ and $x \geqslant \alpha >0$
$$\frac{nx}{e^{nx}} < \frac{n \alpha}{e^{n \alpha}}.$$
Hence, the series converges uniformly on $[\alpha, \infty)$ by the Weierstrass M-test.
