Limit of $s_n=\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}\right)$ \begin{align*}S_n=\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}\right)\end{align*}
how to calculate the limit $s_n$?
\begin{align*}\lim_{n\to \infty } \, S_n\end{align*}

\begin{align}
{1 \over \sqrt{n}}\,\sum_{k = 1}^{n}{1 \over \sqrt{k\,}}
&=
{1 \over \sqrt{n}}\,\sum_{k = 1}^{n}{1 \over \sqrt{n\xi_{k}\,}}\,n\Delta\xi
=
\sum_{k = 1}^{n}{1 \over \sqrt{\xi_{k}\,}}\,\Delta\xi
\sim
\int_{1/n}^{1}{{\rm d}\xi \over \xi^{1/2}}
=
\left.\vphantom{\LARGE A}\;2\xi^{1/2}\right\vert_{1/n}^{1}
\\[3mm]&=
2\left(1 - {1 \over \sqrt{n\,}}\right)
=
2 - {2 \over \sqrt{n\,}} \to \color{#ff0000}{\Large 2}
\quad\mbox{when}\quad
n \to \infty
\end{align}
