I want to prove the following equation: $ \prod_{i = 1}^{n-1} (1 + \frac{1}{i})^i = \frac{n^n}{n!} $
There should be an inductive way and a direct way to prove this for $ n \geq 2 $. I have absolutely no clue how to prove this in a direct manner. However, for the inductive method I got as far as to prove that it's true for $n=2$. But showing that it's true for "n+1" if it's true for any n is a problem for me too. I got as far as that using the binomial theorem:
$ \sum_{i=0}^{n} \frac{n!n^i}{i!(n - i)!} = \sum_{i=0}^{n} \frac{n! \cdot n^n}{i!(n - i)!n^i} $
But I don't see why this equation would be true. If someone could tell me the name of this formula so I can look it up or help me doing the inductive prove I would really appreciate it. Also a hint towards how this could be proved directly would help me a lot. Thanks in advance!