Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$
In the following, I'm going to write what I've done.
By Taylor expansion, $$e^x=\sum_{k=0}^n \frac{x^k}{k!}+\frac{1}{n!}\int_{0}^x (x-t)^ne^tdt.$$
Letting $x=n$,
$$e^n=\sum_{k=0}^n \frac{n^k}{k!}+\frac{1}{n!}\int_{0}^n (n-t)^ne^tdt.$$
Dividing the both sides by $e^n$, $$1=e^{-n}\sum_{k=0}^n \frac{n^k}{k!}+\frac{e^{-n}}{n!}\int_{0}^n (n-t)^ne^tdt.$$
Hence, I know that what I need to do is to find the following limitation: $$\lim_{n\to\infty}\frac{e^{-n}}{n!}\int_{0}^n (n-t)^ne^tdt$$
However, I'm facing difficulty. I need your help.
