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### Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
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### maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$

prove or disprove this $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ this problem is from when Find this limit ...
### Asymptotics of $\sum_{i=0}^{n} \frac{i n^i}{i!}$.
How can you calculate the asymptotics of $\sum_{i=0}^{n} \frac{i n^i}{i!}$ ? This sum looks similar to the power series for $e^n$. I have also seen similar problems solved using Poisson distributions ...