# Arithmetical averages, summation method of

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

$$\sum_{k=0}^\infty u_k.$$

is summable by the method of arithmetical averages to the sum $s$ if

$$\lim_{n\to\infty}\frac{s_0+\dots+s_n}{n+1}=s,$$

where $s_n=\sum\nolimits_{k=1}^nu_k$. In this case, one also says that the sequence $\{s_n\}$ is summable by the method of arithmetical averages to the limit $s$. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. Cesàro summation methods). The summation method of arithmetical averages is completely regular (see Regular summation methods) and translative (see Translativity of a summation method).

#### References

 [1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)

#### Comments

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

#### References

 [a1] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
How to Cite This Entry:
Arithmetical averages, summation method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetical_averages,_summation_method_of&oldid=43581
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article