$$n \in \mathbb{N} $$ $$ \prod_{k=1}^{n-1}\left( 1 + \frac{1}{k}\right)^k = \frac{n^n}{n!}$$
How do I prove this by induction?
I tried something like this:
$$ \left( 1 + \frac{1}{1}\right)^1 \left( 1 + \frac{1}{2}\right)^2 \times \dots \times \left( 1 + \frac{1}{n-1}\right)^{n-1} \left( 1 + \frac{1}{n}\right)^n = \frac{n^n}{n!} \left( 1 + \frac{1}{n}\right)^n $$
And then tried to rearrange the right side until:
$$ \frac{(n+1)^{n+1}}{(n+1)!} $$
But that didn't work. Am I missing something or doing something wrong?
Thanks for the help!
You essentially did it: $$\frac{n^n}{n!}(1+\frac1n)^n=\frac{n^n}{n!}\frac{(n+1)^n}{n^n}= \frac{(n+1)^n}{n!}\frac{n+1}{n+1}=\frac{(n+1)^{n+1}}{(n+1)!}$$