Cesàro summation

Let ${\displaystyle (a_{n})_{n=1}^{\infty }}$ be a sequence, and let

${\displaystyle s_{k}=a_{1}+\cdots +a_{k}=\sum _{n=1}^{k}a_{n}}$

be its kth partial sum.

The sequence (a_n) is called Cesàro summable, with Cesàro sum A ∈ ${\displaystyle \mathbb {R} }$, if, as n tends to infinity, the arithmetic mean of its first n partial sums s₁, s₂, ..., s_n tends to A:

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}s_{k}=A.}$

The value of the resulting limit is called the Cesàro sum of the series ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}.}$ If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.

In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, α) for non-negative integers α. The (C, 0) method is just ordinary summation, and (C, 1) is Cesàro summation as described above.

The higher-order methods can be described as follows: given a series Σa_n, define the quantities

${\displaystyle {\begin{aligned}A_{n}^{-1}&=a_{n}\\A_{n}^{\alpha }&=\sum _{k=0}^{n}A_{k}^{\alpha -1}\end{aligned}}}$

(where the upper indices do not denote exponents) and define E^α
_n to be A^α
_n for the series 1 + 0 + 0 + 0 + .... Then the (C, α) sum of Σa_n is denoted by (C, α)-Σa_n and has the value

${\displaystyle (\mathrm {C} ,\alpha ){\text{-}}\sum _{j=0}^{\infty }a_{j}=\lim _{n\to \infty }{\frac {A_{n}^{\alpha }}{E_{n}^{\alpha }}}}$

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

${\displaystyle (\mathrm {C} ,\alpha ){\text{-}}\sum _{j=0}^{\infty }a_{j}=\lim _{n\to \infty }\sum _{j=0}^{n}{\frac {\binom {n}{j}}{\binom {n+\alpha }{j}}}a_{j}.}$

Even more generally, for α ∈ ${\displaystyle \mathbb {R} }$ \ ${\displaystyle \mathbb {Z} }$⁻, let A^α
_n be implicitly given by the coefficients of the series

${\displaystyle \sum _{n=0}^{\infty }A_{n}^{\alpha }x^{n}={\frac {\displaystyle {\sum _{n=0}^{\infty }a_{n}x^{n}}}{(1-x)^{1+\alpha }}},}$

and E^α
_n as above. In particular, E^α
_n are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σa_n is defined as above.

If Σa_n has a (C, α) sum, then it also has a (C, β) sum for every β > α, and the sums agree; furthermore we have a_n = o(n^α) if α > −1 (see little-o notation).

Let α ≥ 0. The integral ${\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx}$ is (C, α) summable if

${\displaystyle \lim _{\lambda \to \infty }\int _{0}^{\lambda }\left(1-{\frac {x}{\lambda }}\right)^{\alpha }f(x)\,dx}$

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

${\displaystyle \lim _{\lambda \to \infty }{\frac {1}{\lambda }}\int _{0}^{\lambda }\int _{0}^{x}f(y)\,dy\,dx}$

which is the limit of means of the partial integrals.

As is the case with series, if an integral is (C, α) summable for some value of α ≥ 0, then it is also (C, β) summable for all β > α, and the value of the resulting limit is the same.