Find this limit: $$ \lim_{n \to \infty } \frac{1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!}}{e^n} $$
My friend managed to solve this through some probability theory (via Poisson limit theorem and Central limit theorem, but I didn't really understand the proof). He got an answer of $ \frac{1}{2} $. But is it possible to solve this without probability theory, but with calculus?
