derivation of zeroth weighted power mean
Let be positive real numbers, and let be positive real numbers such that . For , the -th weighted power mean of is
Using the Taylor series expansion , where is Landau notation for terms of order and higher, we can write as
By substituting this into the definition of , we get
Again using a Taylor series, this time , we get
Taking the limit , we find
In particular, if we choose all the weights to be ,
the geometric mean of .
Title | derivation of zeroth weighted power mean |
---|---|
Canonical name | DerivationOfZerothWeightedPowerMean |
Date of creation | 2013-03-22 13:10:29 |
Last modified on | 2013-03-22 13:10:29 |
Owner | pbruin (1001) |
Last modified by | pbruin (1001) |
Numerical id | 6 |
Author | pbruin (1001) |
Entry type | Derivation |
Classification | msc 26B99 |
Related topic | PowerMean |
Related topic | GeometricMean |
Related topic | GeneralMeansInequality |
Related topic | DerivationOfHarmonicMeanAsTheLimitOfThePowerMean |