Probabalistic way to calculate limit of a sum

Question

I am following a course on stochastic processes. One question is to Calculate $$lim_{n\to \infty} \sum^n_{i=0} e^{-n} \frac{n^i}{i!}$$ I see that it looks like the expected value of $e^{-n} \frac{n^{i+1}}{i!}$ but i don't know how to proceed. I tried to use mathematica but i'm not sure this is correct.

Solve[Integrate[E^(-n) n^(i + 1)/Factorial[i], {n, 0, x}] == 1/2 Integrate[ E^(-n) n^(i + 1)/Factorial[i], {n, 0, Infinity}] , x, Reals]

Thanks in advance

Answer 1

Hint: Consider a sequence of independent and identically distributed $\mathrm{Pois}(1)$ distributed random variables, and apply the central limit theorem. The value should be $\frac12$.