In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, Cesàro summation (also known as the Cesàro mean
[
][
]) 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).
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
Let
be a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
, and let
:
be its th
partial sum.
The sequence is called Cesàro summable, with Cesàro sum , if, as tends to infinity, the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
of its first ''n'' partial sums tends to :
:
The value of the resulting limit is called the Cesàro sum of the series
If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.
Examples
First example
Let for . That is,
is the sequence
:
Let denote the series
:
The series is known as
Grandi's series.
Let
denote the sequence of partial sums of :
:
This sequence of partial sums does not converge, so the series is divergent. However, Cesàro summable. Let
be the sequence of arithmetic means of the first partial sums:
:
Then
:
and therefore, the Cesàro sum of the series is .
Second example
As another example, let for . That is,
is the sequence
:
Let now denote the series
:
Then the sequence of partial sums
is
:
Since the sequence of partial sums grows without bound, the series diverges to infinity. The sequence of means of partial sums of G is
:
This sequence diverges to infinity as well, so is 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.
summation
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called for non-negative integers . The method is just ordinary summation, and is Cesàro summation as described above.
The higher-order methods can be described as follows: given a series , define the quantities
:
(where the upper indices do not denote exponents) and define to be for the series . Then the sum of is denoted by and has the value
:
if it exists . This description represents an -times iterated application of the initial summation method and can be restated as
:
Even more generally, for , let be implicitly given by the coefficients of the series
:
and as above. In particular, are the
binomial coefficients of power . Then the sum of is defined as above.
If has a sum, then it also has a sum for every , and the sums agree; furthermore we have if (see
little- notation).
Cesàro summability of an integral
Let . The
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
is summable if
:
exists and is finite . The value of this limit, should it exist, is the sum of the integral. Analogously to the case of the sum of a series, if , the result is convergence of the
improper integral. In the case , 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 summable for some value of , then it is also summable for all , and the value of the resulting limit is the same.
See also
*
Abel summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must ...
*
Abel's summation formula
*
Abel–Plana formula
*
Abelian and tauberian theorems
*
Almost convergent sequence A bounded real sequence (x_n) is said to be ''almost convergent'' to L if each Banach limit assigns
the same value L to the sequence (x_n).
Lorentz proved that (x_n) is almost convergent if and only if
:\lim\limits_ \fracp=L
uniformly in n.
The a ...
*
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several va ...
*
Divergent series
*
Euler summation
*
Euler–Boole summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by
: \frac=\sum_^\infty E_n(x)\frac.
The concept is named after Leonhard Euler and George Boole
George Boole (; 2 Novembe ...
*
Fejér's theorem
*
Hölder summation In mathematics, Hölder summation is a method for summing divergent series introduced by .
Definition
Given a series
: a_1+a_2+\cdots,
define
:H^0_n=a_1+a_2+\cdots+a_n
:H^_n=\frac
If the limit
:\lim_H^k_n
exists for some ''k'', this is called ...
*
Lambert summation
*
Perron's formula
*
Ramanujan summation
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has prop ...
*
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a Series (mathematics), series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz me ...
*
Silverman–Toeplitz theorem
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a con ...
*
Stolz–Cesàro theorem
*
Summation by parts
References
Bibliography
*
* . Reprinted 1986 with .
*
*
{{DEFAULTSORT:Cesaro summation
Summability methods
Means