Bound the tail of a convergent geometric series

The series $1+e^{-x}+e^{-2x}+\cdots$ has a sum to infinity. Show that $x>0$. When $x=10$, write $S_n$ and $S$, the finite and infinite sums. Show that $S-S_n<S/10^{100}$ requires $n>10\ln10$.