TITLE: Limit of $n(n^{1\over n}-1)$
QUESTION [6 upvotes]: How can you prove that $a_n=n(n^{1\over n}-1)$ diverges to $\infty $ without L'Hopital's rule. My guess would be by comparison but I can't find any sequence diverging to $\infty$ that's smaller than $a_n$
Any suggestions?

REPLY [6 votes]: We have 
$$n^{1/n}=\exp((\log n)/n)\gt 1+\frac{\log n}{n}$$
if $n\gt 1$. The inequality $\exp(t)\gt 1+t$ for $t\gt 0$ can be proved in many ways.