definition
edit
In mathematical analysis, the concept opposite to convergence is divergence. Divergent Series refers to non-convergent series (in Cauchy sense). Such as series
and
In other words, there is no finite limit for any part or sequence of this sequence.
If a series is convergent, the terms of this series will definitely tend to zero. Therefore, any series whose term does not tend to zero is divergent. However, convergence is a stronger requirement than this: not every series whose term tends to zero converges. One of the counterexamples is harmonic series.
Medieval mathematician oris proved the divergence of harmonic series. [ 1]?
Computable method
edit
In practical mathematical research and the application of physics, astronomy and other disciplines, various divergent series are often naturally involved, so mathematicians try to objectively give this kind of divergent series a real value or a complex value and define it as the sum of the corresponding series, and study the divergent series involved in this sense. Each definition is called summability, and it is also understood as the mapping from series to real or complex numbers, usually linear functional, such as Abel summability, cesaro summability and Borel summability.
The sum formula usually keeps the convergence value of the convergent series, but for some divergent series, this sum formula can additionally define the sum of the corresponding series. For example, cesaro can sum the Grandi series.
Can be summed to 1/2. Most summation methods are related to the analytic extension of the corresponding power series. Every suitable summable method tries to describe the average performance of a sequence when it tends to infinity, and it can also be understood as the average of an infinite sequence. [2]?
history
edit
Before19th century, Euler and other mathematicians widely used divergent series, but it often led to chaotic and contradictory results. The main problem is that Euler thinks that every divergent series should have a natural sum, without defining the meaning of the sum of divergent series in advance. Cauchy finally gave a strict definition of the sum of (convergent) series, and then divergent series was basically excluded from mathematics. It was not until 1886 that they reappeared in Poincare's works on progressive series. In 1890, cesaro realized that he could give a strict definition of the sum of a class of divergent series, and thus defined cesaro sum. (This is not the first time that it has been applied to cesaro and Frobenius in 1880; Cesaro's key contribution is not to discover this kind of summability, but to think that "the precise definition of the sum of divergent series should be given". One year after cesaro's paper was published, other mathematicians have given other definitions of the sum of divergent series, but these definitions are not always compatible: different definitions may give different sums to the same divergent series. Therefore, when referring to the sum of divergent series, it is necessary to specify which summability to use, although most commonly used summability and summability are compatible in a sense. [3]?
Summability theorem of summation of divergent series
edit
When we say that m is regular, we mean that the sum of each convergent series is consistent with its original sum in Cauchy sense. This result is called Abel type theorem of M, which is based on Abel theorem. More interesting, and usually more subtle, is the partial inverse of this result, called Taubel Theorem, which is based on a theorem proved by Taubel. The so-called partial inverse here, exactly, is that if m can sum the series σ and σ satisfies some additional conditions, then σ is inherently convergent. But without any additional conditions, the result is that m is only a summable convergent series (which makes it useless as a divergent series).
The function that a convergent series maps to its sum is linear, so according to Hahn-Barnach theorem, it can be deduced that this function can be expanded into the sum of a series that can be bounded with any part. This fact is generally not very useful, because many of these expansions are incompatible with each other, and because the existence of this operator proves that it appeals to axiom of choice or its equivalent forms, such as Zuo En Lemma, which are unstructured.
Divergent series, as a field of analysis, is essentially concerned with clear and natural techniques, such as Abel and cesaro and borell and other related objects. The appearance of Wiener-Taubel theorem indicates that this branch has entered a new stage, which leads to an unexpected connection between Banach algebra and summability in Fourier analysis.
As a numerical technique, the summation of divergent series is also related to interpolation and sequence transformation. Examples of this technique include Padre approximation, Levin-like sequence transformation and sequence mapping related to renormalization technique of higher-order perturbation theory in quantum mechanics. [4]?
Accessibility in the traditional sense
edit
Conventional convergence and absolute convergence are the sum of two series in the traditional sense, and the discussion here is only for completeness; Strictly speaking, they are not summations of divergent series, because only when these summations fail can we say that a series diverges. Most summations of divergent series are extensions of these two summations on a larger series.
Sum of series
Cauchy's classical definition of series a0+a 1+ sum ... is the limit of partial sum sequence a0+ ...+An. Through the definition of addition operation between two real numbers, and according to mathematical induction, it is not difficult for us to naturally define the addition between finite real numbers. However, the definition of addition between finite real numbers does not mean that we can directly deduce the definition of series sum, because at this time, we have not defined the concept of infinite addition, and we can only clarify the concept of series sum through additional definitions with the help of limit.
Absolute convergence
Given a convergent series A that converges to S, if the term of series A is arbitrarily replaced to get series A', then the convergence of A' always converges to S, then series A is said to be absolutely convergent. Under this definition, it can be proved that a series converges if and only if the new series obtained by taking the absolute values of its terms converges in the classical sense. In some places, the latter is defined as absolute convergence, but the former is more general because it does not involve the concept of absolute value. [5]?
n？ Lund average
edit
abstract
Take the positive term sequence pn from p0 and satisfy.
We use the sequence P to transform the sequence S, and give the weighted average, that is, take
When m tends to infinity, if the limit of tm exists, it is called n? R average or n? Rlund and Np(s), the corresponding summability is called n? Rlund can sum.
n？ The summability of Rlund is completely regular, linear and stable. Surprisingly, any two n? Rlund is compatible with both laws.
Cesaro Kefa
The most special n? Rlund summability is cesaro summability.
Consider series
commemorate
For its partial sum, remember again.
. if
The cesaro sum of this series is called.
. This is obviously an n? Rlund can sum.
As a generalization, let p be
We define N(p)(s) as cesaro and Ck(s), and k is not necessarily an integer. When k≥ 0, cesaro sum is also n? Rlund sum, so it is completely regular, linear, stable and pairwise compatible. Where C0 is the traditional sum and C 1 is the traditional cesaro sum. In addition, if h>k, then Ch is stronger than Ck. [5]?
Abel type summability
edit
Suppose λ= {λ0, λ 1, λ2, ...} is a sequence that strictly increases to infinity, λ0≥ 0. if
For each real number x>0, define its Abel mean \ Abel summability and Aλ as
More generally, if the series f converges to only one big x, but can be analytically extended to every positive real x, then the corresponding summability can still be defined in the above way.
This kind of series is also called generalized Dirichlet series; In physical applications, this is called hot-core regularization method.
Abelian summation is regular, linear, but unstable, and two different Abelian summations are not always compatible. However, some of them are very important.
Abel method
If λn=n, we get Abel sum. and
Where z=exp (? X). So when X right tends to 0, the limit of f(x) is just the limit of power series f(z) when Z left tends to 1. So Abel and A(s) can also be defined as
Abel sum is interesting in a sense, because it is compatible with every cesaro sum and more powerful, that is, there is always A(s) =Ck(s), as long as the latter is defined. Abel sum is regular, linear, stable and compatible with cesaro sum.
Lindevkov method
If λn=nlog(n), we get Lindleff sum (the index starts from 1), and there are
So L(s), or Lindleff sum, is the limit of f(x) when x right tends to 0. Lindefu sum is a very powerful sum. If it is applied to a power series with positive convergence radius, it can be summed everywhere in the Tammy-Leffler star field of this power series.
Accurately speaking, if g(z) is an analytic function that is analyzed at the origin, there will be a corresponding McLaughlin series with positive convergence radius, and there will always be L(G(z)) =g(z) in its Tammy-Leffler star field. In addition, L(G(z)) uniformly converges to G(z) on every compact set of this star domain. [5]?
uniqueness theorem
edit
Some summation methods involve the discussion of analytic continuation of correlation functions.
Analytic continuation of power series
If σσanx converges to a small complex x and can be extended from x=0 to x= 1 along a certain path, the sum of series can be defined as the value of the extension function at x= 1. This value may depend on the choice of path.
Euler method
Euler summability is essentially an exact form of analytic continuation. If a power series converges to a small complex number z, and can be changed from a radius of? The circle of 1/(q+ 1) is analytically extended to a circle with a radius of 1, which is continuous at z= 1, so the value here is called the sum of Euler and ... or (e, q) of series a0+. Euler generally applied this concept before the definition of analytic continuation, and gave the exact form of analytic continuation of power series.
The operation of Euler transformation can be repeated several times, which is essentially equivalent to considering the analytic continuation of power series at z= 1
Analytical continuation of Dirichlet series
Consider Dirichlet series
Analytically extended to the value of s=0, if it exists, it is unique. Defining it as the sum of corresponding series gives a summability. This additivity is sometimes confused with the regularization of zeta function.
Regularization of zeta function
If this series
(for positive an) converges to a large real number s, which can be analytically extended to s=? 1, then it is in s=? The value at 1 is called the zeta regular sum of the sequence a 1+a2+ ... This generalized sum is nonlinear. In applications, ai is sometimes the eigenvalue of a compactly decomposed self-adjoint operator, so f(s) is the trace of A. For example, if A has the eigenvalue 1, 2,3, ... then f(s) is a Riemannian zeta function, and zeta (s) is in s=? What is the value of 1? 1 12, which assigns the corresponding sum to the divergent series1+2+3+4+. Other values at s can also be understood as defining corresponding generalized sums, such as zeta (0) =1+1+1+... =? 12、ζ(? 2) = 1 + 4 + 9 + ...= 0。 Generally speaking,
Where Bk is Bernoulli number. [5]