Cesàro summation

In mathematical analysis, Cesàro summation (also known as the Cesàro mean<ref name="Hardy"> Template:Cite book</ref><ref name="Katznelson"> Template:Cite book </ref> or Cesàro limit<ref>Template:Cite book</ref>) 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 Template:Mvarth partial sum.

The sequence Template:Math is called Cesàro summable, with Cesàro sum Template:Math, if, as Template:Mvar tends to infinity, the arithmetic mean of its first n partial sums Template:Math tends to Template: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 Template:Math for Template:Math. That is, <math>(a_n)_{n=0}^\infty</math> is the sequence

<math>(1, -1, 1, -1, \ldots).</math>

Let Template:Mvar denote the series

<math>G = \sum_{n=0}^\infty a_n = 1-1+1-1+1-\cdots </math>

The series Template:Mvar is known as Grandi's series.

Let <math>(s_k)_{k=0}^\infty</math> denote the sequence of partial sums of Template: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 Template:Mvar is divergent. However, Template:Mvar Template:Em Cesàro summable. Let <math>(t_n)_{n=1}^\infty</math> be the sequence of arithmetic means of the first Template: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 Template:Mvar is Template:Math.

Second example

As another example, let Template:Math for Template:Math. That is, <math>(a_n)_{n=1}^\infty</math> is the sequence

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

Let Template: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 Template:Mvar diverges to infinity. The sequence Template: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 Template:Mvar is Template:Em Cesàro summable. In fact, for the series of any sequence which diverges to (positive or negative) infinity, the Cesàro method also leads to the series of a sequence that diverges likewise, and hence such a series is not Cesàro summable.

Template:Math summation

In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called Template:Math for non-negative integers Template:Mvar. The Template:Math method is just ordinary summation, and Template:Math is Cesàro summation as described above.

The higher-order methods can be described as follows: given a series Template: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 Template:Mvar to be Template:Mvar for the series Template:Nowrap. Then the Template:Math sum of Template:Math is denoted by Template: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 Template:Harv. This description represents an Template:Mvar-times iterated application of the initial summation method and can be restated as

<math>\begin{align}

(\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\\&=\lim_{n\to\infty}\sum_{j=0}^n\frac{\left(n-j+1\right)_\alpha}{\left(n+1\right)_\alpha}a_j\text{.} \end{align}</math>

Even more generally, for Template:Math, let Template: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 Template:Mvar as above. In particular, Template:Mvar are the binomial coefficients of power Template:Math. Then the Template:Math sum of Template:Math is defined as above.

If Template:Math has a Template:Math sum, then it also has a Template:Math sum for every Template:Math, and the sums agree; furthermore we have Template:Math if Template:Math (see [[Big O notation#Little-o notation|little-Template:Mvar notation]]).

Cesàro summability of an integral

Let Template:Math. The integral <math>\textstyle\int_0^\infty f(x)\,dx</math> is Template: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 Template:Harv. The value of this limit, should it exist, is the Template:Math sum of the integral. Analogously to the case of the sum of a series, if Template:Math, the result is convergence of the improper integral. In the case Template:Math, Template: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 Template:Math summable for some value of Template:Math, then it is also Template:Math summable for all Template: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

• Template:Citation
• Template:Citation. Reprinted 1986 with Template:ISBN.
• Template:Springer
• Template:Citation