I'd like to approximate $$\sum_{k=1}^N \frac{1}{k!}$$
I know that $\sum_{n=0}^\infty \frac{1}{n!}=e$ but since I am dealing with only the finite case I'm not sure this approximation is very good. Is there a better one? Note: Not looking for the closed form with the $\Gamma$ function, an approximation in terms of elementary functions is desired.