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Band Of Brothers : L\\\\\\\\\\\\\\\\’Enfer Du Pacifique Saison 1 VF TORRENT

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Is it possible to evaluate $\sum\limits_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}+\sqrt{i}}$

Is it possible to evaluate the following series?
$$\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}+\sqrt{i}}$$
I’m quite sure that it converges but I don’t know how to evaluate it.
Thanks.

A:

I think this is a bit of a loose end (at least at first glance).
The series $\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}+\sqrt{i}}$
can be rewritten as $\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}}-\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i}}$.
The first series diverges by the ratio test (as the terms are decreasing) and the second converges.
But that second series can be rewritten as
$\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}+\sqrt{i}}$ $=\frac{1}{2}\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}}$ $-\frac{1}{2}\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i}}$ $=\frac{1}{2}\left(1-1\right)\sum_{i=1}^{\infty}\frac{\sqrt{i}}{\sqrt{i+1}}$ $

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Band Of Brothers: L\\\\\\\’Enfer Du Pacifique Saison 1 VF Torrent

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