<s>[INST] Determine the convergence of $\sum_{n=1}^\infty (1-cos(\frac{1}{n}))(\sqrt{n+1}-\sqrt{n})$ I'm having trouble determining the convergence of the series:
$\sum_{n=1}^\infty (1-cos(\frac{1}{n}))(\sqrt{n+1}-\sqrt{n})$
I used $\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}}$ and then tried the comparison test for $0 \le (1-cos(\frac{1}{n}))(\sqrt{n+1}-\sqrt{n}) \le (\sqrt{n+1}-\sqrt{n})$ but I can't conclude anything even If I know $\sum_{n=1}^\infty (\frac{1}{\sqrt{n+1}+\sqrt{n}})$ diverges.
I thought about trying the limit test but I couldn't find a sequence to conclude the convergency.
I also know $\sum_{n=1}^\infty 1-cos(\frac{1}{n})$ converges as $1−cos(\frac{1}{n})$ behaves like $\frac{C}{n^2}$ for large values of n
Any guidance, tip or resolution would be highly aprecciated, thank you.
 [/INST] Use :
$$1-\cos(\frac{1}{n}) \sim_{+\infty} \frac{1}{2n^2} $$
and
\begin{align}
 \sqrt{n+1} - \sqrt{n} &= \sqrt{n} (\sqrt{1+\frac{1}{n}} -1) \\
&\sim_{+\infty}  \sqrt{n} \times \frac{1}{2n} \\
&= \frac{1}{2\sqrt{n}}\\
\end{align}
Thus
$$ (1-\cos(\frac{1}{n}))(\sqrt{n+1}-\sqrt{n}) \sim_{+\infty} \frac{1}{4 n^{\frac{5}{2}}}$$
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