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

  • $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ You're welcome, good luck! $\endgroup$ – ArsenBerk Jun 23 at 23:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.