# 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:$$\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n} \sqrt{n+1}} = \frac{1}{\sqrt{n}}-\frac{1}{\sqrt {n+1}}$$
• Thank you very much but how can I get there. Did you multiplied the fraction by $\frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}$ ? – JJ Abrams Jun 23 at 23:03
• No, you don't have to make this too complicated. Just write it as $\frac{\sqrt{n+1}}{\sqrt{n} \sqrt{n+1}} - \frac{\sqrt{n}}{\sqrt{n} \sqrt{n+1}}$. – ArsenBerk Jun 23 at 23:06