Let $(x\_n)\_{n\ge2}$, $x\_2>0$, that satisfies recurrence $x\_{n+1}=\sqrt[n]{1+n x\_n}-1, n\ge 2$. Compute $\lim\_{n\to\infty} nx\_n$.  

It's clear that $x\_n\to 0$, and probably Stolz theorem would be helpful. Is it really necessary to use this theorem?