Given positive numbers a1 < b1, define two sequences recursively by an+1 = anbn, bn+1 =
Chapter 10, Problem 87(choose chapter or problem)
Given positive numbers a1 < b1, define two sequences recursively by an+1 = anbn, bn+1 = an + bn 2 (a) Show that an bn for all n (Figure 14). (b) Show that {an} is increasing and {bn} is decreasing. (c) Show that bn+1 an+1 bn an 2 . (d) Prove that both {an} and {bn} converge and have the same limit. This limit, denoted AGM(a1, b1), is called the arithmetic-geometric mean of a1 and b1. (e) Estimate AGM(1, 2) to three decimal places. an an + 1 bn + 1 bn Geometric mean Arithmetic mean
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