The Arithmetic-Geometric Mean of Gauss

Abstract

The arithmetic-geometric mean of two numbers a and b is defined to be the common limit of the two sequences {a _n } ^∞ _{n = 0} and {b _n } ^∞ _{n = 0} determinec by the algorithm

$$\begin{array}{l} {a_0} = a,\quad \quad \quad \quad \quad \quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_0} = b, \\ {a_{n + 1}} = \left( {{a_n} + {b_n}} \right)/2,\;\quad \quad {b_{n + 1}} = {\left( {{a_n}{b_n}} \right)^{1/2}},\quad n = 0,1,2, \ldots \\ \end{array}$$

((0.1))