# Show a function series converges

• Aug 12th 2009, 03:20 PM
Twig
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
• Aug 12th 2009, 03:40 PM
luobo
Quote:

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.
• Aug 13th 2009, 08:40 AM
Twig
Quote:

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.