calculus - Limit of the sequence $a_n = frac{1+sqrt2+sqrt 3+ cdots +sqrt n}{nsqrt n}$

I have to solve this sequence:

$$a_n = \frac{1+\sqrt2+\sqrt 3+ \cdots +\sqrt n}{n\sqrt n}$$

As a tip I've been told to use Stoltz-Cesaro for sequance of this form : $a_n = \dfrac{x_n}{y_n}$

So I did Stoltz-Cesaro

$\dfrac{x_n - x_{n-1}}{y_n-y_{n-1}}$ and I end up with: $\dfrac{\sqrt{n}}{n\sqrt{n}-(n-1)\sqrt{n-1}}$. I am stuck at this point, can you please give me some tips on what to do next? Thank you.

Answer

A possible approach:

$$\frac{\sqrt n}{n\sqrt n-(n-1)\sqrt{n-1}}=\frac1{n-(n-1)\sqrt{1-\frac1n}}=$$

$$=\frac{n+(n-1)\sqrt{1-\frac1n}}{n^2-(n-1)^2\left(1-\frac1n\right)}=\frac{n+(n-1)\sqrt{1-\frac1n}}{n^2-n^2+n+2n-2-1+\frac1n}=$$

$$\frac{\left(1+\sqrt{1-\frac1n}\right)n-\sqrt{1-\frac1n}}{3n-3+\frac1n}\xrightarrow[n\to\infty]{}\frac23$$