Cesàro summation
Encyclopedia
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.
Cesàro summation is named for the Italian analyst Ernesto Cesàro
(1859–1906).
, and let
be the kth partial sum of the series
The series {sn} is called Cesàro summable, with Cesàro sum
, if the average value of its partial sums tends to 
:
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.
Then the sequence of partial sums {sn} is
so that the series, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {(s1 + ... + sn)/n} are
so that
Therefore the Cesàro sum of the sequence {an} is 1/2.
On the other hand, let an = 1 for n ≥ 1. That is, {an} is the sequence
Then the sequence of partial sums {sn} is
and the series diverges to infinity. The terms of the sequence {(s1 + ... + sn)/n} are
Thus, this sequence also diverges to infinity, and the series is not Cesàro summable. More generally, for a series which diverges to (positive or negative) infinity the Cesàro method leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable. Since a sequence that is ultimately monotonic either converges or diverges to infinity, it follows that a series which is not convergent but Cesàro summable oscillates
.
The higher-order methods can be described as follows: given a series Σan, define the quantities
and define Enα to be Anα for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σan is denoted by (C, α)-Σan and has the value
if it exists . This description represents an
-times iterated application of the initial summation method and can be restated as
Even more generally, for
, let Anα be implicitly given by the coefficients of the series
and Enα as above. In particular, Enα are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σ an is defined as above.
The existence of a (C, α) summation implies every higher order summation, and also that an = o(nα) if α > −1.
is Cesàro summable (C, α) if
exists and is finite . 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.
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.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, Cesàro summation is an alternative means of assigning a sum to an infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
. 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.
Cesàro summation is named for the Italian analyst Ernesto Cesàro
Ernesto Cesàro
Ernesto Cesàro was an Italian mathematician who worked in the field of differential geometry.Cesàro was born in Naples. He is known also for his 'averaging' method for the summation of divergent series, known as the Cesàro mean.-Books by E. Cesaro:* * Ernesto Cesàro (March 12, 1859 – September...
(1859–1906).
Definition
Let {an} be a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
, and let
be the kth partial sum of the series
The series {sn} is called Cesàro summable, with Cesàro sum
, if the average value of its partial sums tends to :In other words, the Cesàro sum of an infinite series is the limit of the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
(average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
) of the first n partial sums of the series, as n goes to infinity.
Examples
Let an = (-1)n+1 for n ≥ 1. That is, {an} is the sequenceThen the sequence of partial sums {sn} is
so that the series, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {(s1 + ... + sn)/n} are
so that
Therefore the Cesàro sum of the sequence {an} is 1/2.
On the other hand, let an = 1 for n ≥ 1. That is, {an} is the sequence
Then the sequence of partial sums {sn} is
and the series diverges to infinity. The terms of the sequence {(s1 + ... + sn)/n} are
Thus, this sequence also diverges to infinity, and the series is not Cesàro summable. More generally, for a series which diverges to (positive or negative) infinity the Cesàro method leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable. Since a sequence that is ultimately monotonic either converges or diverges to infinity, it follows that a series which is not convergent but Cesàro summable oscillates
Oscillation (mathematics)
In mathematics, oscillation is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; that is, oscillation is the failure to have a limit, and is also a quantitative measure for that.Oscillation is defined as the...
.
(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 Σan, define the quantities
and define Enα to be Anα for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σan is denoted by (C, α)-Σan and has the value
if it exists . This description represents an
-times iterated application of the initial summation method and can be restated asEven more generally, for
, let Anα be implicitly given by the coefficients of the seriesand Enα as above. In particular, Enα are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σ an is defined as above.
The existence of a (C, α) summation implies every higher order summation, and also that an = o(nα) if α > −1.
Cesàro summability of an integral
Let α ≥ 0. The integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
is Cesàro summable (C, α) ifexists and is finite . 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
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
. 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.
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
- Borel summation
- Euler summationEuler summationEuler summation is a summability method for convergent and divergent series. Given a series Σan, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series....
- Cesàro mean
- Divergent seriesDivergent seriesIn mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
- Fejér's theoremFejér's theoremIn mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequence of Cesàro means of the sequence of partial sums of the Fourier series of f converges uniformly to f on...
- Riesz meanRiesz meanIn mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...
- Abelian and tauberian theoremsAbelian and tauberian theoremsIn mathematics, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber...
- Silverman–Toeplitz theoremSilverman–Toeplitz theoremIn mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular...
- Summation by parts