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I'm trying to show Convergence or divergence of following series:

$\sum^\infty_{n=1} \sqrt {2 - a_{n}} $

$a_{n+1} = \sqrt {2 + a_{n}}$ , $a_{1} = \sqrt 2$

I know that $a_{n}$ is converge to 2, and therefore $\sqrt {2 - a_{n}}$ converge to 0, so I still can't conclude for series convergence or divergence.

I tried to use direct convergence tests by finding other sequence that bounds $a_{n}$ up or down to conclude for convergence or divergence, but unfortunately without any success so far. (For example I tried to use the series $b_{n} = 2 - \frac 1 {n^4}$ to show series convergence, but I actually realized that $a_{n} < b_{n}$ for n > 1)

Any suggestions or hits will be greatly appreciate!!

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use the ratio test $$\left(\frac{a_{n+1}}{a_n}\right)^2=\frac{2-\sqrt{2+a_n}}{2-a_n}$$ Then rationalize the numerator by multiplying by $2+\sqrt{2+a_n}$

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Let $b_n$=sqrt(2-$a_n$) and then apply the ratio test.

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