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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.

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

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  • $\begingroup$ Thank you very much but how can I get there. Did you multiplied the fraction by $\frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}$ ? $\endgroup$ – JJ Abrams Jun 23 at 23:03
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    $\begingroup$ No, you don't have to make this too complicated. Just write it as $\frac{\sqrt{n+1}}{\sqrt{n} \sqrt{n+1}} - \frac{\sqrt{n}}{\sqrt{n} \sqrt{n+1}}$. $\endgroup$ – ArsenBerk Jun 23 at 23:06
  • $\begingroup$ Oh, I completely forgot about that! Thanks a lot !!!! $\endgroup$ – JJ Abrams Jun 23 at 23:17
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    $\begingroup$ You're welcome, good luck! $\endgroup$ – ArsenBerk Jun 23 at 23:19

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