Editing Generalized mean (section)

==Definition==
If {{mvar|p}} is a non-zero [[real number]], and <math>x_1, \dots, x_n</math> are positive real numbers, then the '''generalized mean''' or '''power mean''' with exponent {{mvar|p}} of these positive real numbers is<ref name="Bullen1"/><ref name = "dC2016">{{cite journal|last=de Carvalho|first=Miguel|title=Mean, what do you Mean?|journal=[[The American Statistician]]|year=2016|volume=70|issue=3|pages=764‒776|doi=10.1080/00031305.2016.1148632|url=https://zenodo.org/record/895400|hdl=20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c|hdl-access=free}}</ref>

<math display=block>M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} .</math>

(See [[Norm (mathematics)#p-norm|{{mvar|p}}-norm]]). For {{math|1=''p'' = 0}} we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

<math display="block">M_0(x_1, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} .</math>

Furthermore, for a [[sequence]] of positive weights {{mvar|w<sub>i</sub>}} we define the '''weighted power mean''' as<ref name="Bullen1"/>
<math display=block>M_p(x_1,\dots,x_n) = \left(\frac{\sum_{i=1}^n w_i x_i^p}{\sum_{i=1}^n w_i} \right)^{{1}/{p}}</math>
and when {{math|1=''p'' = 0}}, it is equal to the [[weighted geometric mean]]:

<math display=block>M_0(x_1,\dots,x_n) = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} .</math>

The unweighted means correspond to setting all {{math|1=''w<sub>i</sub>'' = 1}}.