derivation of geometric mean as the limit of the power mean

Fix x1,x2,,xn+. Then let


For r0, by definition μ(r) is the rth power meanMathworldPlanetmath of the xi. It is also clear that μ(r) is a differentiable function for r0. What is limr0μ(r)?

We will first calculate limr0logμ(r) using l’Hôpital’s rule (

limr0logμ(r) =limr0log(x1r++xnrn)r

It follows immediately that

Title derivation of geometric meanMathworldPlanetmath as the limit of the power mean
Canonical name DerivationOfGeometricMeanAsTheLimitOfThePowerMean
Date of creation 2013-03-22 14:17:13
Last modified on 2013-03-22 14:17:13
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Derivation
Classification msc 26D15
Related topic LHpitalsRule
Related topic PowerMean
Related topic WeightedPowerMean
Related topic ArithmeticGeometricMeansInequality
Related topic ArithmeticMean
Related topic GeometricMean
Related topic DerivationOfZerothWeightedPowerMean