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!

Thanks in advance!



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