# Cesàro summation

# Cesàro summation

In mathematical analysis, **Cesàro summation** (also known as the **Cesàro mean**^{[1]}^{[2]}) assigns values to some infinite sums that are not 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.

Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906). It is a special case of a matrix summability method.

The term *summation* can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the *sum* of that series is 1/2.

Definition

`Letbe asequence, and let`

be its kth partial sum.

The sequence (*a**n*) is called **Cesàro summable**, with Cesàro sum *A* ∈ ℝ, if, as n tends to infinity, the arithmetic mean of its first *n* partial sums *s*1, *s*2, ..., *s**n* tends to A:

`The value of the resulting limit is called the Cesàro sum of the seriesIf this series is (conditionally) convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.`

Examples

First example

`Let`

*a*_{n}= (−1)^{n}for*n*≥ 0. That is,is the sequenceLet G denote the series

The series G is known as Grandi's series.

`Letdenote the sequence of partial sums ofG:`

`This sequence of partial sums does not converge, so the seriesGis divergent. However,G`

*is*Cesàro summable. Letbe the sequence of arithmetic means of the firstnpartial sums:Then

and therefore, the Cesàro sum of the series G is 1/2.

Second example

`As another example, let`

*a*_{n}=*n*for*n*≥ 1. That is,is the sequenceLet G now denote the series

`Then the sequence of partial sumsis`

Since the sequence of partial sums grows without bound, the series G diverges to infinity. The sequence (*t**n*) of means of partial sums of G is

This sequence diverges to infinity as well, so G is *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

*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 Eαn to be Aαn 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 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 ∑*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 *an* = *o*(*nα*) if *α* > −1 (see little-o notation).

Cesàro summability of an integral

`Let`

*α*≥ 0. Theintegralis(C,*α*)summable ifexists 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.

See also

Abel summation

Abel's summation formula

Abel–Plana formula

Abelian and tauberian theorems

Almost convergent sequence

Borel summation

Divergent series

Euler summation

Euler–Boole summation

Fejér's theorem

Hölder summation

Lambert summation

Ramanujan summation

Riesz mean

Silverman–Toeplitz theorem

Stolz–Cesàro theorem

Summation by parts

## References

*Divergent Series*. Providence: American Mathematical Society. ISBN 978-0-8218-2649-2.

*An Introduction to Harmonic Analysis*. New York: Dover Publications. ISBN 978-0-486-63331-2.