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?

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