Arithmetic-Harmonic Mean

Let

$\displaystyle a_{n+1}$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(a_n+b_n)$	(1)
$\displaystyle b_{n+1}$	$\textstyle =$	$\displaystyle {2a_nb_n\over a_n+b_n}.$	(2)

Then

$\begin{displaymath} A(a_0, b_0)=\lim_{n\to\infty} a_n=\lim_{n\to\infty} b_n =\sqrt{a_0b_0}, \end{displaymath}$

(3)

which is just the Geometric Mean.