Problem:
Prove that the series $f\left ( x \right )= \sum_{n=0}^{\infty } e^{-n x}$ converges for all $x>0$ and that $f$ is infinitely differentiable on $(0, \infty)$.
My solution is: I treated the sum as a sum of geometric series $f\left ( x \right )= \sum_{n=0}^{\infty } \left ( e^{-x } \right )^{n}$ and then $f(x)= \frac{1}{1- e^{-x}}$. Does this prove the convergence of f for all $x\in \mathbb{R}$?
Now I am stuck on how to prove that $f$ is infinitely differentiable. Can you guys give me a detailed proof? Thanks