harmonic mean

If $a_{1},\,a_{2},\,\ldots,\,a_{n}$ 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}}+\cdots+\frac{1}{a_{n}}}

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It follows easily the estimation

$H.M.\;<\;na_{i}\quad(i\;=\;1,\,2,\,\ldots,\,n).$
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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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If one draws through the intersecting point of the diagonals of a trapezoid a line parallel to the parallel sides of the trapezoid, then the segment of the line inside the trapezoid is equal to the harmonic mean of the parallel sides.
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In the harmonic series

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$

every following it.

Title	harmonic mean
Canonical name	HarmonicMean
Date of creation	2013-03-22 11:50:50
Last modified on	2013-03-22 11:50:50
Owner	drini (3)
Last modified by	drini (3)
Numerical id	15
Author	drini (3)
Entry type	Definition
Classification	msc 11-00
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