In an assignment for the course Real-Analysis I have to show that for all $n \ge 1$ applies: $$a_n = \displaystyle \sum_{k=0}^n \frac1{k!}\left(1-\frac1n\right)\left(1-\frac2n\right)\cdots\left(1-\frac{k-1}n\right)$$ where $a_n$ is given as: $$a_n =\left(1+\frac1n\right)^n$$
I have to use the binomial formula for this: $(x+y)^n=\displaystyle\sum_{k=0}^n \binom{n}{k} x^{k} y^{n-k}$
However, me and my fellow students have been trying to figure it out for some time now, but we can't...