arithmetic-geometric mean as a product

Recall that, given two real numbers 0<xy, their arithmetic-geometric meanDlmfDlmfMathworldPlanetmath may be defined as M(x,y)=limngn, where

g0 =x
a0 =y
gn+1 =angn
an+1 =an+gn2.

In this entry, we will re-express this quantity as an infinite product. We begin by rewriting the recursion for gn:


From this, it follows that


where hn=an/gn.

As it stands, this is not so interesting because no way has been given to determine the factors hn other than first computing an and gn. We shall now correct this defect by deriving a recursion which may be used to compute the hn’s directly:

hn+1 =an+1gn+1

Taking the limit n, we then have the formulaMathworldPlanetmathPlanetmath





Title arithmetic-geometric mean as a productPlanetmathPlanetmath
Canonical name ArithmeticgeometricMeanAsAProduct
Date of creation 2013-03-22 17:09:59
Last modified on 2013-03-22 17:09:59
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Derivation
Classification msc 26E60
Classification msc 33E05