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Difference between revisions of "Harmonic mean"

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''of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464901.png" />''
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''of numbers $a_1,\dots,a_n$''
  
 
The number reciprocal to the [[Arithmetic mean|arithmetic mean]] of the reciprocals of the given numbers, i.e. the number
 
The number reciprocal to the [[Arithmetic mean|arithmetic mean]] of the reciprocals of the given numbers, i.e. the number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464902.png" /></td> </tr></table>
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$$\frac{n}{\frac{1}{a_1}+\dots+\frac{1}{a_n}}.$$
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464903.png" /> is the harmonic mean of the fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046490/h0464906.png" />. The harmonic mean of given numbers is never greater than their arithmetic mean.
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Thus, $1/n$ is the harmonic mean of the fractions $1/(n-1)$ and $1/(n+1)$, $n=2,3,\dots$. The harmonic mean of given numbers is never greater than their arithmetic mean.

Latest revision as of 13:55, 30 December 2018

of numbers $a_1,\dots,a_n$

The number reciprocal to the arithmetic mean of the reciprocals of the given numbers, i.e. the number

$$\frac{n}{\frac{1}{a_1}+\dots+\frac{1}{a_n}}.$$

Thus, $1/n$ is the harmonic mean of the fractions $1/(n-1)$ and $1/(n+1)$, $n=2,3,\dots$. The harmonic mean of given numbers is never greater than their arithmetic mean.

How to Cite This Entry:
Harmonic mean. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_mean&oldid=17187
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article