Cesàro summation

Definition

Let {a_n} be a sequence, and let

be the kth partial sum of the series

The series ∑ ∞
n = 1 a_n is called Cesàro summable, with Cesàro sum A ∈ ℝ, if the average value of its partial sums s_k tends to A:

In other words, the Cesàro sum of an infinite series is the limit of the arithmetic mean (average) of the first n partial sums of the series, as n goes to infinity. If a series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum. For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum.

Examples

Let a_n = (−1)ⁿ for n ≥ 0. That is, {a_n} is the sequence

Let G denote the series

This series G is known as Grandi's series.

Let {s_k} denote the sequence of partial sums, up to k:

This does not converge to a single number. Let {t_n} denote the sequence

This results in

Therefore the Cesàro sum of the series G is 1 / 2.

Lets try another example, where a_n = n for n ≥ 1. That is, {a_n} is the sequence

and let G now denote the series

Then the sequence of partial sums {s_n} is

and the evaluation of G diverges to infinity. The terms of the sequence of means of partial sums {t_n} are here

Thus, this sequence diverges to infinity as well as G, and G is now not Cesàro summable. In fact, for any sequence which diverges to (positive or negative) infinity the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable.

(C, α) summation

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 ∑a_n, define the quantities

(where the upper indices do not denote exponents) and define Eα
n to be Aα
n for the series 1 + 0 + 0 + 0 + …. Then the (C, α) sum of ∑a_n is denoted by (C, α)-∑a_n 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 ∈ ℝ ℤ⁻, let Aα
n be implicitly given by the coefficients of the series

and Eα
n as above. In particular, Eα
n are the binomial coefficients of power −1 − α. Then the (C, α) sum of ∑a_n is defined as above.

If ∑a_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).

Cesàro summability of an integral

Let α ≥ 0. The integral ∫∞
0 f(x) dx is Cesàro summable (C, α) 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.

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.