Show $1+2^2+...+n^n$ ≤ $(1+1/n-1)*n^n$

Question

$1+2^2+...+n^n$ ≤ $(1+1/(n-1))*n^n$

Well what I come up with is it's left to prove $1+2^2+...+n^{n-1}$≤ $n^n/(n-1)$

I think I need to somehow come up with a summation of the former then compare it with $n^n/(n-1)$.

Answer 1

$$ n^n=n^n-1+1=(n-1)(n^{n-1}+n^{n-2}+\dots+1)+1. $$