Find the value of : $\lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotso +\frac{1}{\sqrt{n+n}} \right)$ I am trying to evaluate $\lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotso +\frac{1}{\sqrt{n+n}} \right)$. I suspect identifying an appropriate Riemman sum is the trick. However after some toying with it I gave up on this suspicion and stumbled across the Stolz-Cesaro theorem, which I then used to calculate the limit as $\sqrt{2}$. 
Does anybody see a way to do this as Riemann sum?
I tried putting it in this form
$\frac{1}{n} \sum_{k=1}^n \sqrt{\frac{n}{n+k}}$ but then I don't see how carry on to identify the function from the partition to integrate. 
Thank you for suggestions or comments.

HINT:
$$\frac{1}{\sqrt{n}}\sum_{k=1}^n\frac{1}{\sqrt{n+k}}=\frac1n\sum_{k=1}^n\frac{1}{\sqrt{1+k/n}}$$
