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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 Limit using Poisson distribution for example. However I cannot see how to handle the extra term.

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$$\sum_{i=0}^{n} \frac{i n^i}{i!} = \sum_{i=1}^{n} \frac{n^i}{(i-1)!} = \sum_{i=0}^{n-1} \frac{ n^{i+1}}{i!} = n\sum_{i=0}^{n-1} \frac{ n^{i}}{i!} $$

Since, as shown in the linked problem, $$e^{-n}\sum_{k=0}^n\frac{n^k}{k!} =\frac12+O(n^{-1/2}) $$

$$\sum_{i=0}^{n} \frac{i n^i}{i!} = n(\sum_{i=0}^{n} \frac{ n^{i}}{i!}-\frac{ n^{n}}{n!}), $$

so

$$\begin{align} e^{-n}\sum_{i=0}^{n} \frac{i n^i}{i!} &= e^{-n}n(\sum_{i=0}^{n} \frac{ n^{i}}{i!}-\frac{ n^{n}}{n!})\\ &=n(1/2 +O(n^{-1/2})) - n e^{-n}\frac{ n^{n}}{n!}\\ &= n/2 + O(n^{1/2}) - n \frac{(n/e)^n}{\sqrt{\pi}(1+O(1/n)) n^{n+1/2}/e^n}\\ &= n/2 + O(n^{1/2}) - n (n/e)^n/(\sqrt{\pi}(1+O(1/n)) n^{n+1/2}/e^n)\\ &= n/2 + O(n^{1/2}) - n^{1/2}(1+O(1/n))/\sqrt{\pi}\\ &= n/2 + O(n^{1/2}) \end{align} $$

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