In mathematical analysis, Cesàro summation (also known as the Cesàro mean[1][2] or Cesàro limit[3]) assigns values to some infinite sums that are not necessarily convergent in the usual sense.
The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
This special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859–1906).
be a sequence, and let be its kth partial sum.
, if, as n tends to infinity, the arithmetic mean of its first n partial sums s1, s2, ..., sn tends to A: The value of the resulting limit is called the Cesàro sum of the series
denote the sequence of partial sums of G: This sequence of partial sums does not converge, so the series G is divergent.
is Since the sequence of partial sums grows without bound, the series G diverges to infinity.
The sequence (tn) of means of partial sums of G is This sequence diverges to infinity as well, so G is not Cesàro summable.
In fact, for the series of any sequence which diverges to (positive or negative) infinity, the Cesàro method also leads to the series of a sequence that diverges likewise, and hence such a series is not Cesàro summable.
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, α) for non-negative integers α.
The higher-order methods can be described as follows: given a series Σan, define the quantities (where the upper indices do not denote exponents) and define Eαn to be Aαn for the series 1 + 0 + 0 + 0 + .... Then the (C, α) sum of Σan is denoted by (C, α)-Σan and has the value if it exists (Shawyer & Watson 1994, pp.16-17).
This description represents an α-times iterated application of the initial summation method and can be restated as Even more generally, for α ∈
−, let Aαn be implicitly given by the coefficients of the series and Eαn as above.
In particular, Eαn are the binomial coefficients of power −1 − α.
Then the (C, α) sum of Σan is defined as above.
is (C, α) summable if exists and is finite (Titchmarsh 1948, §1.15).
The value of this limit, should it exist, is the (C, α) sum of the integral.
Analogously to the case of the sum of a series, if α = 0, the result is convergence of the improper integral.
In the case α = 1, (C, 1) convergence is equivalent to the existence of the limit which is the limit of means of the partial integrals.