Cesàro summation

Let $(a\_n)\_\{n=1\}^\backslash infty$ be a sequence, and let

:$s\_k\; =\; a\_1\; +\; \backslash cdots\; +\; a\_k=\; \backslash sum\_\{n=1\}^k\; a\_n$

be its {{mvar|k}}th partial sum.

The sequence {{math|(a_n)}} is called Cesàro summable, with Cesàro sum {{math|A ∈ $\backslash mathbb\{R\}$}}, if, as {{mvar|n}} tends to infinity, the arithmetic mean of its first n partial sums {{math|s₁, s₂, ..., s_n}} tends to {{mvar|A}}:

:$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{1\}\{n\}\backslash sum\_\{k=1\}^n\; s\_k\; =\; A.$

The value of the resulting limit is called the Cesàro sum of the series $\backslash textstyle\backslash sum\_\{n=1\}^\backslash infty\; a\_n.$ If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.

Let {{math|a_n {{=}} (−1)ⁿ}} for {{math|n ≥ 0}}. That is, $(a\_n)\_\{n=0\}^\backslash infty$ is the sequence

:$(1,\; -1,\; 1,\; -1,\; \backslash ldots).$

Let {{mvar|G}} denote the series

:$G\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; =\; 1-1+1-1+1-\backslash cdots$

The series {{mvar|G}} is known as Grandi's series.

Let $(s\_k)\_\{k=0\}^\backslash infty$ denote the sequence of partial sums of {{mvar|G}}:

:$\backslash begin\{align\}$

s_k &= \sum_{n=0}^k a_n \\

(s_k) &= (1, 0, 1, 0, \ldots).

\end{align}

This sequence of partial sums does not converge, so the series {{mvar|G}} is divergent. However, {{mvar|G}} {{em|is}} Cesàro summable. Let $(t\_n)\_\{n=1\}^\backslash infty$ be the sequence of arithmetic means of the first {{mvar|n}} partial sums:

:$\backslash begin\{align\}$

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}

Then

:$\backslash lim\_\{n\backslash to\backslash infty\}\; t\_n\; =\; 1/2,$

and therefore, the Cesàro sum of the series {{mvar|G}} is {{math|1/2}}.

As another example, let {{math|a_n {{=}} n}} for {{math|n ≥ 1}}. That is, $(a\_n)\_\{n=1\}^\backslash infty$ is the sequence

:$(1,\; 2,\; 3,\; 4,\; \backslash ldots).$

Let {{mvar|G}} now denote the series

:$G\; =\; \backslash sum\_\{n=1\}^\backslash infty\; a\_n\; =\; 1+2+3+4+\backslash cdots$

Then the sequence of partial sums $(s\_k)\_\{k=1\}^\backslash infty$ is

:$(1,\; 3,\; 6,\; 10,\; \backslash ldots).$

Since the sequence of partial sums grows without bound, the series {{mvar|G}} diverges to infinity. The sequence {{math|(t_n)}} of means of partial sums of G is

:$\backslash left(\backslash frac\{1\}\{1\},\; \backslash frac\{4\}\{2\},\; \backslash frac\{10\}\{3\},\; \backslash frac\{20\}\{4\},\; \backslash ldots\backslash right).$

This sequence diverges to infinity as well, so {{mvar|G}} is {{em|not}} 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.

In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called {{math|(C, α)}} for non-negative integers {{mvar|α}}. The {{math|(C, 0)}} method is just ordinary summation, and {{math|(C, 1)}} is Cesàro summation as described above.

The higher-order methods can be described as follows: given a series {{math|Σa_n}}, define the quantities

:

(where the upper indices do not denote exponents) and define {{mvar|E{{su|b=n|p=α}}}} to be {{mvar|A{{su|b=n|p=α}}}} for the series {{nowrap|1 + 0 + 0 + 0 + ...}}. Then the {{math|(C, α)}} sum of {{math|Σa_n}} is denoted by {{math|(C, α)-Σa_n}} and has the value

:$(\backslash mathrm\{C\},\backslash alpha)\backslash text\{-\}\backslash sum\_\{j=0\}^\backslash infty\; a\_j=\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{A\_n^\backslash alpha\}\{E\_n^\backslash alpha\}$

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

:$\backslash begin\{align\}$

(\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}

Even more generally, for {{math|α ∈ $\backslash mathbb\{R\}$ \ $\backslash mathbb\{Z\}$⁻}}, let {{mvar|A{{su|b=n|p=α}}}} be implicitly given by the coefficients of the series

:$\backslash sum\_\{n=0\}^\backslash infty\; A\_n^\backslash alpha\; x^n=\backslash frac\{\backslash displaystyle\{\backslash sum\_\{n=0\}^\backslash infty\; a\_nx^n\}\}\{(1-x)^\{1+\backslash alpha\}\},$

and {{mvar|E{{su|b=n|p=α}}}} as above. In particular, {{mvar|E{{su|b=n|p=α}}}} are the binomial coefficients of power {{math|−1 − α}}. Then the {{math|(C, α)}} sum of {{math|Σa_n}} is defined as above.

If {{math|Σa_n}} has a {{math|(C, α)}} sum, then it also has a {{math|(C, β)}} sum for every {{math|β > α}}, and the sums agree; furthermore we have {{math|a_n {{=}} o(n^α)}} if {{math|α > −1}} (see Big O notation#Little-o notation).

Let {{math|α ≥ 0}}. The integral $\backslash textstyle\backslash int\_0^\backslash infty\; f(x)\backslash ,dx$ is {{math|(C, α)}} summable if

:$\backslash lim\_\{\backslash lambda\backslash to\backslash infty\}\backslash int\_0^\backslash lambda\backslash left(1-\backslash frac\{x\}\{\backslash lambda\}\backslash right)^\backslash alpha\; f(x)\backslash ,\; dx$

exists and is finite {{harv|Titchmarsh|1948|loc=§1.15}}. The value of this limit, should it exist, is the {{math|(C, α)}} sum of the integral. Analogously to the case of the sum of a series, if {{math|α {{=}} 0}}, the result is convergence of the improper integral. In the case {{math|α {{=}} 1}}, {{math|(C, 1)}} convergence is equivalent to the existence of the limit

:$\backslash lim\_\{\backslash lambda\backslash to\; \backslash infty\}\backslash frac\{1\}\{\backslash lambda\}\backslash int\_0^\backslash lambda\; \backslash int\_0^x\; f(y)\backslash ,\; dy\backslash ,dx$

which is the limit of means of the partial integrals.

As is the case with series, if an integral is {{math|(C, α)}} summable for some value of {{math|α ≥ 0}}, then it is also {{math|(C, β)}} summable for all {{math|β > α}}, and the value of the resulting limit is the same.

{{div col|colwidth=20em}}

• 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

{{div col end}}

{{Reflist}}

• {{citation |last1=Shawyer|first1=Bruce|last2=Watson |first2=Bruce|title=Borel's Methods of Summability: Theory and Applications |year=1994 |publisher=Oxford University Press |isbn=0-19-853585-6}}
• {{citation|last=Titchmarsh|first=E. C.|author-link=Edward Charles Titchmarsh|title=Introduction to the theory of Fourier integrals|edition=2nd|year=1948|publisher=Chelsea Publishing|location=New York, NY}}. Reprinted 1986 with {{ISBN|978-0-8284-0324-5}}.
• {{springer|title=Cesàro summation methods|first=I. I.|last=Volkov|year=2001|id=c/c021360}}
• {{citation|last=Zygmund|first=Antoni|author-link=Antoni Zygmund|title= Trigonometric Series |title-link= Trigonometric Series |edition=2nd|year=1988|orig-year=1968|publisher=Cambridge University Press|isbn=978-0-521-35885-9}}

{{DEFAULTSORT:Cesaro summation}}

Category:Summability methods

Category:Means