Approximating finite sum of factorial reciprocal

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.

There is an exact representation $$\sum_{k=1}^n \frac{1}{k!}=e\frac{ \Gamma (n+1,1)}{\Gamma (n+1)}-1=\frac{\lfloor e \,n!\rfloor }{n!}-1$$

If you use Taylor's formula, you get $$ \sum_{k=1}^N \frac{1}{k!} = e - 1 -\frac{e^{\xi}}{(N+1)!}, \quad \xi \in (0,1) $$

So, if you choose to approximate the sum by $e-1$, the error you commit is bounded by $\frac{e}{(N+1)!}$. As an example, if $N=10$ you get an error of the order $10^{-8}$.