Английская Википедия:Cesàro summation
Шаблон:Short description Шаблон:For
In mathematical analysis, Cesàro summation (also known as the Cesàro mean[1][2] or Cesàro limit[3]) 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 <math>(a_n)_{n=1}^\infty</math> be a sequence, and let
- <math>s_k = a_1 + \cdots + a_k= \sum_{n=1}^k a_n</math>
be its Шаблон:Mvarth partial sum.
The sequence Шаблон:Math is called Cesàro summable, with Cesàro sum Шаблон:Math, if, as Шаблон:Mvar tends to infinity, the arithmetic mean of its first n partial sums Шаблон:Math tends to Шаблон:Mvar:
- <math>\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k = A.</math>
The value of the resulting limit is called the Cesàro sum of the series <math>\textstyle\sum_{n=1}^\infty a_n.</math> If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.
Examples
First example
Let Шаблон:Math for Шаблон:Math. That is, <math>(a_n)_{n=0}^\infty</math> is the sequence
- <math>(1, -1, 1, -1, \ldots).</math>
Let Шаблон:Mvar denote the series
- <math>G = \sum_{n=0}^\infty a_n = 1-1+1-1+1-\cdots </math>
The series Шаблон:Mvar is known as Grandi's series.
Let <math>(s_k)_{k=0}^\infty</math> denote the sequence of partial sums of Шаблон:Mvar:
- <math>\begin{align}
s_k &= \sum_{n=0}^k a_n \\
(s_k) &= (1, 0, 1, 0, \ldots).
\end{align}</math>
This sequence of partial sums does not converge, so the series Шаблон:Mvar is divergent. However, Шаблон:Mvar Шаблон:Em Cesàro summable. Let <math>(t_n)_{n=1}^\infty</math> be the sequence of arithmetic means of the first Шаблон:Mvar partial sums:
- <math>\begin{align}
t_n &= \frac{1}{n}\sum_{k=0}^{n-1} s_k \\
(t_n) &= \left(\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\right).
\end{align}</math>
Then
- <math>\lim_{n\to\infty} t_n = 1/2,</math>
and therefore, the Cesàro sum of the series Шаблон:Mvar is Шаблон:Math.
Second example
As another example, let Шаблон:Math for Шаблон:Math. That is, <math>(a_n)_{n=1}^\infty</math> is the sequence
- <math>(1, 2, 3, 4, \ldots).</math>
Let Шаблон:Mvar now denote the series
- <math>G = \sum_{n=1}^\infty a_n = 1+2+3+4+\cdots </math>
Then the sequence of partial sums <math>(s_k)_{k=1}^\infty</math> is
- <math>(1, 3, 6, 10, \ldots).</math>
Since the sequence of partial sums grows without bound, the series Шаблон:Mvar diverges to infinity. The sequence Шаблон:Math of means of partial sums of G is
- <math>\left(\frac{1}{1}, \frac{4}{2}, \frac{10}{3}, \frac{20}{4}, \ldots\right).</math>
This sequence diverges to infinity as well, so Шаблон:Mvar is Шаблон:Em 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.
Шаблон:Math summation
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called Шаблон:Math for non-negative integers Шаблон:Mvar. The Шаблон:Math method is just ordinary summation, and Шаблон:Math is Cesàro summation as described above.
The higher-order methods can be described as follows: given a series Шаблон:Math, define the quantities
- <math>\begin{align} A_n^{-1}&=a_n \\ A_n^\alpha&=\sum_{k=0}^n A_k^{\alpha-1} \end{align}</math>
(where the upper indices do not denote exponents) and define Шаблон:Mvar to be Шаблон:Mvar for the series Шаблон:Nowrap. Then the Шаблон:Math sum of Шаблон:Math is denoted by Шаблон:Math and has the value
- <math>(\mathrm{C},\alpha)\text{-}\sum_{j=0}^\infty a_j=\lim_{n\to\infty}\frac{A_n^\alpha}{E_n^\alpha}</math>
if it exists Шаблон:Harv. This description represents an Шаблон:Mvar-times iterated application of the initial summation method and can be restated as
- <math>(\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.</math>
Even more generally, for Шаблон:Math, let Шаблон:Mvar 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 Шаблон:Mvar as above. In particular, Шаблон:Mvar are the binomial coefficients of power Шаблон:Math. Then the Шаблон:Math sum of Шаблон:Math is defined as above.
If Шаблон:Math has a Шаблон:Math sum, then it also has a Шаблон:Math sum for every Шаблон:Math, and the sums agree; furthermore we have Шаблон:Math if Шаблон:Math (see [[Big O notation#Little-o notation|little-Шаблон:Mvar notation]]).
Cesàro summability of an integral
Let Шаблон:Math. The integral <math>\textstyle\int_0^\infty f(x)\,dx</math> is Шаблон:Math summable if
- <math>\lim_{\lambda\to\infty}\int_0^\lambda\left(1-\frac{x}{\lambda}\right)^\alpha f(x)\, dx </math>
exists and is finite Шаблон:Harv. The value of this limit, should it exist, is the Шаблон:Math sum of the integral. Analogously to the case of the sum of a series, if Шаблон:Math, the result is convergence of the improper integral. In the case Шаблон:Math, Шаблон:Math convergence is equivalent to the existence of the limit
- <math>\lim_{\lambda\to \infty}\frac{1}{\lambda}\int_0^\lambda \int_0^x f(y)\, dy\,dx</math>
which is the limit of means of the partial integrals.
As is the case with series, if an integral is Шаблон:Math summable for some value of Шаблон:Math, then it is also Шаблон:Math summable for all Шаблон:Math, 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
- Perron's formula
- Ramanujan summation
- Riesz mean
- Silverman–Toeplitz theorem
- Stolz–Cesàro theorem
- Cauchy's limit theorem
- Summation by parts
References
Bibliography
- Шаблон:Citation
- Шаблон:Citation. Reprinted 1986 with Шаблон:ISBN.
- Шаблон:Springer
- Шаблон:Citation
