How to show the sum of (n+1)^n/n^(n+1) diverges

http://www.wolframalpha.com/input/?i=sum%28%28n%2B1%29%5En%2Fn%5E%28n%2B1%29%29+

Note that we can rewrite the n-th term as [(n+1)^n / n^n] * (1/n) = (1 + 1/n)^n * (1/n),
and that lim(n→∞) (1 + 1/n)^n = e.

This gives me the idea to use the Limit Comparison Test with the harmonic series.
lim(n→∞) [(1 + 1/n)^n * (1/n)] / (1/n)
= lim(n→∞) (1 + 1/n)^n
= e.

Since the harmonic series diverges, so must the sum in question.

I hope this helps!