Putnam 2006 B6

Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define $$a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}$$ for $n > 0$. Evaluate $$\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.$$