# Proof of convergence with central limit theorem [duplicate]

I'm trying to solve some problems and this one showed up: Show that $$e^{-n} \left(1 + n + \frac{n^2}{2!} + \dots + \frac{n^n}{n!}\right) \to \frac{1}{2}$$ as $$n \to \infty$$. The problem sugests to apply the Central Limit Theorem to a sequence of random variables with Poisson distribution. Can somebody help me with this? I don't know how to apply it correctly. And if someone knows some another way to prove this, I'll apreciate!