Calculate the sum of series with square roots Calculate the sum of the following series using partial sums:
$$\sum_{n=1}^\infty \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n} \sqrt{n+1}} $$
I rationalized the upper part of the fraction but I got lost. Could you please help me showing the steps of the how to transform the fraction into a partial sum? Thanks in advance.  

HINT:$$\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n} \sqrt{n+1}} = \frac{1}{\sqrt{n}}-\frac{1}{\sqrt {n+1}}$$
