# Convergence or divergence of series with recursive sequence

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!!

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