Prove that
{1+(1/1)}^1 x {1+(1/2)}^2 ...{1+[1/(n-1)]}^(n-1) = {n^(n-1)}/{(n-1)!}
I assume this has to be done by induction, but I'm having trouble working it out.
There's actually no need for induction.
$\displaystyle \displaystyle \prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^k=\prod_{k=1}^{n-1}\left(\frac{k+1}{k}\right)^k$
$\displaystyle =\dfrac{2\cdot3^2\cdot4^2\cdots n^{n-1}}{1\cdot2^2\cdot3^3\cdots (n-1)^{n-1}}$
Notice all the cancellations...