limits, series, derangements: simplifying the limit of a series to 1/e

I am trying to find the limit as n goes to infinity of the probability that a random permutation "deranges" a sequence of integers.

i started by writing a formula for the number of derangements that are possible in the sequence. then i divided that by the total number of combinations ($n!$) to find the probability. now i want the limit of the probability, so i took the limit as n is going to infinity. now, i have the
$$\lim_{n\to\infty}\sum_{k=0}^n\frac{(-1)^k}{k!}$$
at this point, i'm a little stuck. i know the answer is supposed to be $1/e$, but im not sure how to get there. is $(-1)^k/k!$ some famous oscillating series, that we can just say goes to the inverse of e. it seems difficult to reduce $k$ to $e$.

Answer

Hint$1$ : For all real $x$, $e^x$can be expressed as $$\sum_{k=0}^\infty \frac{x^k}{k!}$$

Hint$2$ : $\frac{1}{e}=e^{-1}$