In mathematics, a divergent series is an infinite series that does not converge. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a series is a sum of a sequence of terms. ...

If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. The simplest example of a divergent series whose terms do approach zero is the harmonic series See harmonic series (music) for the (related) musical concept. ...

The divergence of the harmonic series was elegantly proven (here) by the medieval mathematician Nicole Oresme. See harmonic series (music) for the (related) musical concept. ... Nicolas Oresme (c. ...

Divergent series can sometimes be assigned a value by using a summability method or a summation method. For example, Cesàro summation assigns Grandi's divergent series In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ... The infinite series 1 − 1 + 1 − 1 + · · · is sometimes called Grandis series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. ...

the value ½. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time and explaining them using mathematics. ... In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance in space which is useful if the divergences arise from short-distance physical effects). ...

Theorems on methods for summing divergent series

For convergent series, a good summability method M agrees with the actual limit of the series. Such a result is called an abelian theorem for M, because the prototype was Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems because of a prototype proved by Alfred Tauber. Here partial converse means that if M sums the series Σ, and some side-condition holds, then Σ was convergent in the first place; without any side condition such a result would say that M only summed convergent series (making it an essentially useless summation method). In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a large number of methods have been proposed for the summation of divergent series. ... In real analysis, Abels theorem for power series relates a limit of a power series to the sum of its coefficients. ... In mathematics, a large number of methods have been proposed for the summation of divergent series. ... Alfred Tauber (1866-1942) was a mathematician. ...

The operator giving the sum of a convergent series is linear, and it follows from the Hahn-Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This fact is not very useful in practice since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma. They are therefore nonconstructive. In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ...

The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and natural techniques such as Abel summation, Cesàro summation and Borel summation, and their relationships. The advent of Wiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebra methods in Fourier analysis. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ... In mathematics, a Borel summation is a generalisation of the usual notion of summation of a series. ... In mathematics, Wieners tauberian theorem is a 1932 result of Norbert Wiener. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. Examples for such techniques are Padé approximants and Levin-type sequence transformations. In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. ... To evaluate the limit of a slowly convergent sequence or series, or the antilimit of a divergent series numerically, one may use extrapolation methods or sequence transformations : For a given series , the transformed sequence is , where the members of the transformed sequence are usually computed from some finite number of... Padé approximant is the best approximation of a function by a rational function of given order. ...

Properties of summation methods

If A is any function assigning a value to a sequence, there are certain properties it is desirable for it to possess if it is to be a useful summation method.

1. Regularity. A method is regular if, whenever the sequence s converges to x, A(s) = x.
2. Linearity. A is linear if it is a linear functional on convergent sequences, so that A(r + s) = A(r) + A(s) and A(ks) = k.A(s), for k a scalar (real or complex.)
3. Stability. If s is a sequence starting from s₀ and s′ is the sequence obtained by truncating the first value, and so starting at s₁, then A(s) is defined if and only if A(s′) is defined, and A(s) = A(s′ ).

The third condition is less important, and some significant methods, such as Borel summation, do not possess it. In mathematics, a Borel summation is a generalisation of the usual notion of summation of a series. ...

A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, A(s) = B(s). If two methods are consistent, and one sums more series than the other, the one summing more series is stronger.

It should be noted that there are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants. To evaluate the limit of a slowly convergent sequence or series, or the antilimit of a divergent series numerically, one may use extrapolation methods or sequence transformations : For a given series , the transformed sequence is , where the members of the transformed sequence are usually computed from some finite number of... Padé approximant is the best approximation of a function by a rational function of given order. ...

Nõrlund means

Suppose p_n is a sequence of positive terms, starting from p₀. Suppose also that

If now we transform a sequence s by using p to give weighted means, setting

then the limit of t_n as n goes to infinity is the associated Nõrlund mean N_p(s).

The Nõrlund mean is regular, linear, and stable. Moreover, any two Nõrlund means which sum the same sequence are consistent. The most significant of the Nõrlund means are the Cesàro sums. Here, if we set

then the Cesàro sum C_k is defined by C_k(s) = N_p^k(s). Cesàro sums are Nõrlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C₀ is ordinary summation, and C₁ is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then C_h is stronger than C_k. In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ...

Abelian means

Suppose λ_n is a strictly increasing sequence tending towards ∞, and that λ₀ ≥ 0. Suppose a_n=s_n+1-s_n is an infinite series, with corresponding sequence s. Suppose

converges for all positive real numbers x. Then the Abelian mean A_λ is defined as

A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heat-kernel regularization. In mathematics and theoretical physics, zeta-function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums. ...

Abelian means are regular, linear, and stable, but not always consistent between one and another. However, some special cases are very important summation methods.

Abel summation

If λ_n = n, then we obtain the method of Abel summation. Here

where z = exp(-x). Then the limit of f(x) as x approaches 0 through positive reals is the limit of the power series for f(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as

Abel summation is interesting in part because it is consistent with but more powerful than Cesàro summation; if C_k(s) = a for any positive k, then A(s) = a. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.

Lindelöf summation

If λ_n = n ln(n), then (indexing from one) we have

Then L(s), the Lindelöf sum, is the limit of f(x) as x goes to zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the Mittag-Leffler star.

If g(z) is analytic in a disk around zero, and hence has a Maclaurin series with G(z) with a positive radius of convergence, then L(G(z)) = g(z) in the Mittag-Leffler star. This is defined by taking rays from the origin out to any singularity, and removing the singularity and anything beyond it on the ray from the complex plane. L(G(z)) therefore extends the definition of G(z) as far as it can be extended without running into the possibility (if the singularity is a branch point) of multiple values. As the degree of the taylor series rises, it approaches the correct function. ...

References

• Divergent Series by G. H. Hardy, Oxford, Clarendon Press, 1949.
• Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.
• Padé Approximants by G. A. Baker, Jr. and P. Graves-Morris, Cambridge U.P., 1996.