Hi!

Problem:Show that $\displaystyle \sum_{n=1}^{\infty}xe^{-nx} $ is convergent for $\displaystyle x\geq 0 $ .Also show that the convergence is uniform on every interval $\displaystyle \left(a,\infty\right) $ , where $\displaystyle a>0$ .

Is the convergence uniform on $\displaystyle \left(0,\infty\right) $ ?

I would appreciate a fairly detalied solution, explaining most of the steps.

Thx