power meanPowerMean2001-10-17 00:18:062004-02-01 20:43:40Definition\usepackage{amssymb}
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\usepackage{xypic}The $r$-th power mean of the numbers $x_1,x_2,\ldots,x_n$ is defined as:
$$M^r(x_1,x_2,\ldots,x_n)=\left(\frac{x_1^r+x_2^r+\cdots+x_n^r}{n}\right)^{1/r}.$$
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The arithmetic mean is a special case when $r=1$.
The power mean is a continuous function of $r$, and taking limit when $r\to0$ gives us the geometric mean:
$$M^0(x_1,x_2,\ldots,x_n)=\sqrt[n]{x_1 {x_{2}} \cdots x_n}.$$
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Also, when $r=-1$ we get
$$M^{-1}(x_1,x_2,\ldots,x_n)=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}$$
the harmonic mean.
A generalization of power means are weighted power means.