convergence of arithmetic-geometric mean

In this entry, we show that the arithmetic-geometric meanDlmfDlmfMathworldPlanetmath converges. By the arithmetic-geometric means inequality, we know that the sequencesMathworldPlanetmath of arithmeticPlanetmathPlanetmath and geometric meansMathworldPlanetmath are both monotonic and boundedPlanetmathPlanetmath, so they converge individually. What still needs to be shown is that they converge to the same limit.

Define xn=an/gn. By the arithmetic-geometric inequalityMathworldPlanetmath, we have xn1. By the defining recursions, we have


Since xn1, we have 1/xn1, and xnxn, hence


From this inequality


we may conclude that xn1 as n, which , by the definition of xn, is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to


Not only have we proven that the arithmetic-geometric mean converges, but we can infer a rate of convergence from our proof. Namely, we have that 0xn-1(x0-1)/2n. Hence, we see that the rate of convergence of an and gn to the answer goes as O(2-n).

By more carefully bounding the recursion for xn above, we may obtain better estimates of the rate of convergence. We will now derive an inequality. Suppose that y0.

0 y5+y4+4y3+3y2
y2+4y+4 y5+y4+4y3+4y2+4y+4
(y+2)2 (y+1)(y2+2)2

Set x=y+1 (so we have x1).

(x+1)2 x((x-1)2+2)2
x x2((x-1)2+2)2(x+1)2
x x((x-1)2+2)x+1
x+1xx (x-1)2+2
12(x+1x) 1+12(x-1)2

Thus, because xn+1=(xn+1/xn)/2, we have


From this equation, we may derive the bound


This is a much better bound! It approaches zero far more rapidly than any exponential functionDlmfDlmfMathworldPlanetmath, so we have superlinear convergence.

Title convergence of arithmetic-geometric mean
Canonical name ConvergenceOfArithmeticgeometricMean
Date of creation 2013-03-22 17:09:46
Last modified on 2013-03-22 17:09:46
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type TheoremMathworldPlanetmath
Classification msc 33E05
Classification msc 26E60