limit of a two variables recursive series

I am looking at a sequence which is defined as the following:
$$a_0 = a$$ $$b_0 = b$$ $$a_{n+1}=\frac{a_n+b_n}{2}$$ $$b_{n+1}=\sqrt{a_nb_n}$$
I know that both series have $a_n\geq a_{n+1} \geq b_{n+1} \geq b_n$ for every $n \geq 1$ and therefore are monotone, bounded, and converge to the same limit. My question - given $a,\,b$ what is the limit?

• Don't confuse series with sequences. – ajotatxe Mar 20 '15 at 17:09

Both sequences converge to the arithmetic-geometric mean of $a$ and $b$.