An approximation to the arithmetic-geometric mean

Abstract

Given positive numbers a > b, consider the ‘agm iteration’ given by a0 = a, b0 = b andAt each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an) and (bn) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).