TITLE: Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$.
QUESTION [5 upvotes]: I have a problem with the calculation of the following limit.
\begin{equation}
\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}
\end{equation}
I do not know where to start!
Thank you very much

REPLY [9 votes]: There are at least two possibilities. In all of them you use that $\lim_{n\to\infty}=\sqrt[n]{n}=1$.
The first one uses the following result: if $a_n$ is a convergent sequence, then
$$\lim_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n}=\lim_{n\to\infty}a_n.$$
The second is to use Stolz's criterion.