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.