Express sequence in closed type Given the sequence $a_n = \sqrt{2+a_{n-1}}$. Is there anyway to find a closed form for this sequence? 
Thank you for your time.

We are given
$a_n = \sqrt{2+a_{n-1}}$.
I will assime that
 $0 \le a_0 < 2$.
If $a_{n-1} < 2$,
$a_n < \sqrt{2+2} = 2$,
so all subsequent $a_n < 2$.
Since $a_1 = \sqrt{2+a_0} < 2$,
all  $a_n < 2$.
Let $d_n = 2-a_n$,
so $a_n = 2-d_n$.
$2-d_n = \sqrt{2+(2-d_{n-1})}
=\sqrt{4-d_{n-1}}
$
or,
since $1-x < \sqrt{1-x} < 1-x/2$
if $0 < x < 1$,
$d_n 
= 2-\sqrt{4-d_{n-1}}
= 2-2\sqrt{1-d_{n-1}/4}
< 2-2(1-d_{n-1}/4)
= d_{n-1}/2
$
so
$d_n \to 0$
geometrically,
so $a_n \to 2$.
