Prove that as (n ightarrow infty), [ e^{-n} sum_{k=0}^{n} frac{n^{k}}{k!} ightarrow frac{1}{2} ] Apply the

Prove that as \(n \rightarrow \infty\),

\[ e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k!} \rightarrow \frac{1}{2} \]

Apply the Lyapunov theorem to the sum of the Poisson distributed random variables with parameter \(\lambda=1\).