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单词 DerivationOfGeometricMeanAsTheLimitOfThePowerMean
释义

derivation of geometric mean as the limit of the power mean


Fix x1,x2,,xn+. Then let

μ(r):=(x1r++xnrn)1/r.

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 (http://planetmath.org/LHpitalsRule).

limr0logμ(r)=limr0log(x1r++xnrn)r
=limr0(x1rlogx1++xnrlogxnn)(x1r++xnrn)
=limr0x1rlogx1++xnrlogxnx1r++xnr
=logx1++logxnn
=logx1xnn.

It follows immediately that

limr0(x1r++xnrn)1/r=x1xnn.
Titlederivation of geometric meanMathworldPlanetmath as the limit of the power mean
Canonical nameDerivationOfGeometricMeanAsTheLimitOfThePowerMean
Date of creation2013-03-22 14:17:13
Last modified on2013-03-22 14:17:13
OwnerMathprof (13753)
Last modified byMathprof (13753)
Numerical id8
AuthorMathprof (13753)
Entry typeDerivation
Classificationmsc 26D15
Related topicLHpitalsRule
Related topicPowerMean
Related topicWeightedPowerMean
Related topicArithmeticGeometricMeansInequality
Related topicArithmeticMean
Related topicGeometricMean
Related topicDerivationOfZerothWeightedPowerMean
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