harmonic mean
If ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ are positive numbers, we define their harmonic mean^{} as the inverse number of the arithmetic mean^{} of their inverse numbers:
$$H.M.=\frac{n}{\frac{1}{{a}_{1}}+\frac{1}{{a}_{2}}+\mathrm{\cdots}+\frac{1}{{a}_{n}}}$$ 

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It follows easily the estimation
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If you travel from city $A$ to city $B$ at $x$ miles per hour, and then you travel back at $y$ miles per hour. What was the average velocity for the whole trip?
The harmonic mean of $x$ and $y$. That is, the average velocity is$$\frac{2}{\frac{1}{x}+\frac{1}{y}}=\frac{2xy}{x+y}.$$  •

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In the harmonic series
$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{\dots}$$ every following it.
Title  harmonic mean 
Canonical name  HarmonicMean 
Date of creation  20130322 11:50:50 
Last modified on  20130322 11:50:50 
Owner  drini (3) 
Last modified by  drini (3) 
Numerical id  15 
Author  drini (3) 
Entry type  Definition 
Classification  msc 1100 
Related topic  ArithmeticMean 
Related topic  GeneralMeansInequality 
Related topic  WeightedPowerMean 
Related topic  PowerMean 
Related topic  ArithmeticGeometricMeansInequality 
Related topic  RootMeanSquare3 
Related topic  ProofOfGeneralMeansInequality 
Related topic  ProofOfArithmeticGeometricHarmonicMeansInequality 
Related topic  HarmonicMeanInTrapezoid 
Related topic  ContraharmonicMean 
Related topic  Contr 