Proof of Divergence of a Series

[Arkuski]

Prove that the series \displaystyle\sum_{k=1}^{\infty}\sqrt[k]{k+1}-1 diverges.

I thought that I could show the n^{th} term was greater than \frac{1}{n} but this is turning out to be more difficult than I imagined. Is there a neat proof that n^n>(n+1)^{n-1}?

 

[Dick]

$(k+1)^\frac{1}{k}=\exp(\log((k+1)^\frac{1}{k}))$. Simplify the log a little and think about what the series expansion of $e^x$ looks like.