# Arithmetical averages, summation method of

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 if

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

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://www.encyclopediaofmath.org/index.php?title=Arithmetical_averages,_summation_method_of&oldid=31461