When trying to answer this question: Find minimum $n$ such that $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$ has no answer inside the circle of radius $100$ centered at the origin
I ended up in what seems to be a dead end, and I'd really like to see it finished. The problem is to find for which $n$, the following inequality holds: $$ \frac{1}{n!}\int_0^{100} x^n e^{-x}dx < e^{-200}, $$ and it would be even better if there was some way to solve the general inequality, $$ \frac{1}{n!}\int_0^{r} x^n e^{-x}dx < e^{-2r}. $$
