View source for Arithmetical averages, summation method of

One of the methods for summing series and sequences. The series

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is summable by the method of arithmetical averages to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134002.png" /> if

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134004.png" />. In this case, one also says that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134005.png" /> is summable by the method of arithmetical averages to the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134006.png" />. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. [[Cesàro summation methods|Cesàro summation methods]]). The summation method of arithmetical averages is completely regular (see [[Regular summation methods|Regular summation methods]]) and translative (see [[Translativity of a summation method|Translativity of a summation method]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>



====Comments====
Instead of  "arithmetical averages"  the term  "summation method of arithmetical meansarithmetical means"  is sometimes used, cf. [[#References|[a1]]], and instead of  "summation"  one also uses  "summability" : summability method.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>