# Compute the sum of the following series

$$\sum_{j=0}^{\infty} \left(e^{-jx}\right)\left(x^j\right)$$for $x$ on $[0, \infty)$.

I already proved that this series converges uniformly for $x$ on $[0, \infty)$, i.e. its partial sum converges uniformly to a function on $[0, \infty)$. But I just could not get a fair guess of what that limiting function is, or how to compute it.

Thanks a lot!

• Hint: Geometric series. – Daniel Fischer Nov 23 '13 at 22:51
• Got it. Thanks a lot! – mflowww Nov 23 '13 at 22:59

This is an infinite series: $1 + \left(e^{-x}x\right) + (e^{-x}x)^2 + (e^{-x}x)^3 + \cdots = (1- e^{-x}x)^{-1}$