Check the uniform convergence of $\ \sum_{n=1}^\infty x^ne^{-xn} $

Question

I have a problem with a power series:

$$\ \sum_{n=1}^\infty x^ne^{-xn} ~~~~ x\in (0, \infty)$$

Could anyone explain how to check uniform, almost uniform and pointwise convergence? Not only proving convergence is problematic for me, but also exponential function makes me confused.

Answer 1

Hint : Let's consider $\displaystyle \sum_{n\le m} \left(\frac{x}{e^{x}}\right)^{n} = \frac{e^x - x(xe^{-x})^m}{e^x - x}$.

Then your series is limit(with respect to $m$) of following partial sum. When does it converge? When does the limit exist?