# Thread: Show a function series converges

1. ## Show a function series converges

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

2. Originally Posted by Twig
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
The series is essentially a geometric series.

3. Originally Posted by luobo
The series is essentially a geometric series.
With $\displaystyle e^{x}$ .

So I get $\displaystyle x \cdot \sum_{n=1}^{\infty}\frac{1}{\left(e^{x}\right)^{n} }$

Alright, thx.