TITLE: Calculate sum of series $\sum \frac{n^2}{n!}$
QUESTION [5 upvotes]: I have to calculate sum of series $\sum \frac{n^2}{n!}$. I know that $\sum \frac{1}{n!}=e$ but I dont know how can I use that fact here..

REPLY [2 votes]: Hint:
Let $f(x)=\displaystyle \sum_{n=0}^{\infty} \frac{x^n}{n!}$.
Express
$
g(x)=\displaystyle \sum_{n=0}^{\infty} \frac{n^2x^{n}}{n!}
$
in terms of $f'(x)$ and $f''(x)$.  
Interpret the result using $f(x)=e^x$.