If t is a nonzero real number, we can define the generalized mean with exponent t of the positive real numbers a_{1},...,a_{n} as
The case t = 1 yields the arithmetic mean and the case t = 1 yields the harmonic mean. As t approaches 0, the limit of M(t) is the geometric mean of the given numbers, and so it makes sense to define M(0) to be the geometric mean. Furthermore, as t approaches ∞, M(t) approaches the maximum of the given numbers, and as t approaches ∞, M(t) approaches the minimum of the given numbers.
In general, if ∞ <= s < t <= ∞, then
See also: average
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