That result isn't correct. For n=5 the sum equals approximately 0.81, and your right hand side equals approximately 0.293

You could have checked that yourself.

Hello,

Attention!If the limits of the sum are irrationals and \(\displaystyle \sqrt{2} \leq k \leq \sqrt{5}\) , then how does the value of \(\displaystyle k\) vary?

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Four questions:

1) How many terms does he have \(\displaystyle \sum_{k=\sqrt{2}}^{\sqrt{2}} \frac{1}{k(k+1)}\)?

2) How many terms does he have \(\displaystyle \sum_{k=\sqrt{2}}^{\sqrt{3}} \frac{1}{k(k+1)}\)?

3) How many terms does he have \(\displaystyle \sum_{k=\sqrt{2}}^{\sqrt{4}} \frac{1}{k(k+1)}\)?

4) How many terms does he have \(\displaystyle \sum_{k=\sqrt{2}}^{\sqrt{5}} \frac{1}{k(k+1)}\)?

Thank you very much!

All the best,

Integrator