Using the binomial theorem, prove the following equation using induction. Please have a look at my current approach, as I welcome feedback on tackling this equation. For, I'm currently experiencing hardships on where to go next.
$$\left(1+\frac{1}{n}\right)^n = 1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]$$
What I have done: starting with induction
$$\left(1+\frac{1}{n}\right)^{n+1} = \left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)^n$$ $$\implies \left(1+\frac{1}{n}\right)\left(1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]\right)$$ $$\implies 1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right] + \frac{1}{n}+ \frac{\sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]}{n}$$
I'm not sure on how to break this down next, please help me.