Cesàro summation

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In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the 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. However, a series which postively diverges to an infinite value, will not be summable to a finite value in any case.

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

Contents 1 Definition 2 Examples 3 (C, α) summation 4 Cesàro summability of an integral 5 See also 6 References

[edit] Definition

Let {a_n} be a sequence, and let

be the kth partial sum of the series

.

The sequence {a_n} is called Cesàro summable, with Cesàro sum A, if

.

[edit] Examples

Let a_n = (-1)ⁿ⁺¹ for n ≥ 1. That is, {a_n} is the sequence

.

Then the sequence of partial sums {s_n} is

,

so that the series, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {(s₁ + ... + s_n)/n} are

,

so that

.

Therefore the Cesàro sum of the sequence {a_n} is 1/2.

[edit] (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 Σa_n, define the quantities

and define E_n^α to be A_n^α for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σa_n is

if it exists (Shawyer & Watson 1994, pp.16-17).

Even more generally, for , let A_n^α be implicitly given by the coefficients of the series

and E_n^α as above. In particular, E_n^α are the binomial coefficients of power -1-α. Then the (C, α) sum of Σa_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.

[edit] Cesàro summability of an integral

Let α ≥ 0. The integral is Cesàro summable (C, α) if

exists and is finite (Titchmarsh 1948, §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

which is the limit of means of the partial integrals.

[edit] See also

• Abel summation
• Borel summation
• Cesàro mean
• Divergent series
• Fejér's theorem
• Riesz mean

[edit] References

• Shawyer, Bruce & Watson, Bruce (1994), Borel's Methods of Summability: Theory and Applications, Oxford UP, ISBN 0-19-853585-6 .
• Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.) (published 1986), ISBN 978-0828403245 .
• Volkov, I.I. (2001), “Cesàro summation methods”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0521358859 .