## Limit of $n!(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!)$ as $n \to \infty$

$$L = \lim_{ n \to \infty} n!(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!) =0 \:$$

I know that $$\lim_{ n \to \infty}\sum_{k=0}^{n} 1/k! =e$$

So I’m assuming that $$L$$ goes to $$0$$ because $$(1-\frac{1}{e}\sum_{k=0}^{n} 1/k!)$$ goes to $$0$$ faster than $$n!$$ goes to infinity. But how to prove this?