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2014-02-02T01:46:24Z

151.40.126.195: Corrected the definition

{{For|the song "Cesaro Summability" by the band Tool|Ænima}}

In [[mathematical analysis]], '''Cesàro summation''' is an alternative means of assigning a sum to an [[Series (mathematics)|infinite series]]. If the series [[Convergent series|converges]] in the usual sense to a sum ''A'', then the series is also Cesàro summable and has Cesàro sum ''A''. The significance of Cesàro summation is that a series which does not converge may still have a well-defined Cesàro sum.

Cesàro summation is named for the Italian analyst [[Ernesto Cesàro]] (1859–1906).

== Definition ==
Let {''a''n} be a [[sequence]], and let

:<math>s_k = a_1 + \cdots + a_k</math>

be the ''k''th [[partial sum]] of the [[Series (mathematics)|series]]

:<math>\sum_{n=1}^\infty a_n.</math>

The series <math>\sum_{n=1}^\infty a_n</math> is called '''Cesàro summable''', with Cesàro sum <math>A \in \R</math>, if the average value of its partial sums <math>s_k</math> tends to <math>A</math>:

:<math>\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k = A.</math>

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. It is easy to show that any [[Convergent Series|convergent series]] is Cesaro summable, and the sum of the series agrees with its Cesaro sum. However, as the first example below demonstrates, there are series that diverge but are nonetheless Cesaro summable.

== Examples ==

Let ''a''''n'' = (−1)''n''+1 for ''n'' ≥ 1. That is, {''a''''n''} is the sequence

:<math>1, -1, 1, -1, \ldots.\,</math>

and let ''G'' denote the series <math> \sum_{n=1}^\infty a_n =1-1+1-1+1-\cdots </math>

Then the sequence of partial sums {''s''''n''} <math> = \sum_{k=1}^n a_k </math> is

:<math>1, 0, 1, 0, \ldots,\,</math>

so that the series ''G'', known as [[Grandi's series]], clearly does not converge. On the other hand, the terms of the sequence {''t''''n''} of the (partial) means of the {''s''''n''} where
:<math> t_n = \frac{1}{n}\sum_{k=1}^n s_k </math>
are

:<math>\frac{1}{1}, \,\frac{1}{2}, \,\frac{2}{3}, \,\frac{2}{4}, \,\frac{3}{5}, \,\frac{3}{6}, \,\frac{4}{7}, \,\frac{4}{8}, \,\ldots,</math>

so that

:<math>\lim_{n\to\infty} t_n = 1/2.</math>

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

On the other hand, now let ''a''''n'' = ''n'' for ''n'' ≥ 1. That is, {''a''n} is the sequence

:<math>1, 2, 3, 4, \ldots.\,</math>

and let ''G'' now denote the series <math> \sum_{n=1}^\infty a_n =1+2+3+4+5+\cdots </math>

Then the sequence of partial sums {''s''n} is

:<math>1, 3, 6, 10, \ldots,\,</math>

and the evaluation of ''G'' diverges to infinity.
The terms of the sequence of means of partial sums {''t''n } are here

:<math>\frac{1}{1}, \,\frac{4}{2}, \,\frac{10}{3}, \,\frac{20}{4}, \,\ldots.</math>

Thus, this sequence diverges to infinity as well as ''G'', and ''G'' is now '''not''' Cesàro summable. In fact, any series 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, ''n'') for non-negative integers ''n''. 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 Σ''a''''n'', define the quantities

:<math>A_n^{-1}=a_n;\quad A_n^\alpha=\sum_{k=0}^n A_k^{\alpha-1}</math>

and define ''E''''n''α to be ''A''''n''α for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σ''a''''n'' is denoted by (C, α)-Σ''a''''n'' and has the value

:<math>(C,\alpha)-\sum_{j=0}^\infty a_j=\lim_{n\to\infty}\frac{A_n^\alpha}{E_n^\alpha}</math>

if it exists {{harv|Shawyer|Watson|1994|loc=pp.16-17}}. This description represents an <math>\alpha</math>-times iterated application of the initial summation method and can be restated as

:<math>(C,\alpha)-\sum_{j=0}^\infty a_j = \lim_{n\to\infty} \sum_{j=0}^n \frac{{n \choose j}}{{n+\alpha \choose j}} a_j.</math>

Even more generally, for <math>\alpha\in\mathbb{R}\setminus(-\mathbb{N})</math>, let ''A''''n''α be implicitly given by the coefficients of the series

:<math>\sum_{n=0}^\infty A_n^\alpha x^n=\frac{\displaystyle{\sum_{n=0}^\infty a_nx^n}}{(1-x)^{1+\alpha}},</math>

and ''E''''n''α as above. In particular, ''E''''n''α are the [[binomial coefficient#Newton's binomial series|binomial coefficients]] of power −1 − α. Then the (C, α) sum of Σ ''a''''n'' is defined as above.

The existence of a (C, α) summation implies every higher order summation, and also that ''a''''n'' = ''o''(''n''α) if α > −1.

== Cesàro summability of an integral ==
Let α ≥ 0. The [[integral]] <math>\scriptstyle{\int_0^\infty f(x)\,dx}</math> is Cesàro summable (C, α) if

:<math>\lim_{\lambda\to\infty}\int_0^\lambda\left(1-\frac{x}{\lambda}\right)^\alpha f(x)\, dx </math>

exists and is finite {{harv|Titchmarsh|1948|loc=§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

:<math>\lim_{\lambda\to \infty}\frac{1}{\lambda}\int_0^\lambda\left\{\int_0^xf(y)\, dy\right\}\,dx</math>

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]]
* [[Borel summation]]
* [[Euler summation]]
* [[Lambert summation]]
* [[Cesàro mean]]
* [[Divergent series]]
* [[Fejér's theorem]]
* [[Riesz mean]]
* [[Perron's formula]]
* [[Abelian and tauberian theorems]]
* [[Silverman–Toeplitz theorem]]
* [[Summation by parts]]

==References==
*{{citation |last1=Shawyer|first1=Bruce|first2=Bruce|last2=Watson |title=Borel's Methods of Summability: Theory and Applications |publisher=Oxford UP |year=1994 |id=ISBN 0-19-853585-6}}.
* {{citation|last=Titchmarsh|first=E|authorlink=Edward Charles Titchmarsh|title=Introduction to the theory of Fourier integrals|isbn=978-0-8284-0324-5|year=1948|edition=2nd|publication-date=1986|publisher=Chelsea Pub. Co.|location=New York, N.Y.}}.
* {{springer|title=Cesàro summation methods|first=I.I.|last=Volkov|year=2001|id=c/c021360}}
* {{citation|title=Trigonometric series|first=Antoni|last=Zygmund|authorlink=Antoni Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.

{{DEFAULTSORT:Cesaro summation}}

[[Category:Summability methods]]

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