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In 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 finite limit.
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. However, convergence is a stronger condition: not all series whose terms approach zero converge. The simplest counterexample is the harmonic series
The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.
In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges. A summability method or summation method is a partial function from the set of sequences of partial sums of series to values. For example, Cesàro summation assigns Grandi's divergent series
the value ^{1}/_{2}. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.
A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an abelian theorem for M, from the prototypical Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems, from a prototype proved by Alfred Tauber. Here partial converse means that if M sums the series Σ, and some sidecondition 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 useless as a summation method for divergent series).
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.
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.
Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. Examples for such techniques are Padé approximants, Levintype sequence transformations, and orderdependent mappings related to renormalization techniques for largeorder perturbation theory in quantum mechanics.
Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluating a = a_{0} + a_{1} + a_{2} + ..., we work with the sequence s, where s_{0} = a_{0} and s_{n+1} = s_{n} + a_{n+1}. In the convergent case, the sequence s approaches the limit a. A summation method can be seen as a function from a set of sequences of partial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanically translate this to a seriessummation method A^{Σ} that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.
The third condition is less important, and some significant methods, such as Borel summation, do not possess it.^{[citation needed]}
One can also give a weaker alternative to the last condition.
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.
There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levintype sequence transformations and Padé approximants, as well as the orderdependent mappings of perturbative series based on renormalization techniques.
Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. For instance, whenever r ≠ 1, the geometric series
can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when r is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of ∞.
Suppose p_{n} is a sequence of positive terms, starting from p_{0}. 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 an average called the Nørlund mean N_{p}(s).
The Nørlund mean is regular, linear, and stable. Moreover, any two Nørlund means are consistent. The most significant of the Nørlund means are the Cesàro sums. Here, if we define the sequence p^{k} by
then the Cesàro sum C_{k} is defined by C_{k}(s) = N_{(pk)}(s). Cesàro sums are Nørlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C_{0} is ordinary summation, and C_{1} is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then C_{h} is stronger than C_{k}.
Suppose λ = {λ_{0}, λ_{1}, λ_{2}, ...} is a strictly increasing sequence tending towards infinity, and that λ_{0} ≥ 0. Suppose
converges for all real numbers x>0. Then the Abelian mean A_{λ} is defined as
More generally, if the series for f only converges for large x but can be analytically continued to all positive real x, then one can still define the sum of the divergent series by the limit above.
A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heatkernel regularization.
Abelian means are regular and linear, but not stable and not always consistent between different choices of λ. However, some special cases are very important summation methods.
If λ_{n} = n, then we obtain the method of Abel summation. Here
where z = exp(−x). Then the limit of ƒ(x) as x approaches 0 through positive reals is the limit of the power series for ƒ(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: A(s) = C_{k}(s) whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.
If λ_{n} = n log(n), then (indexing from one) we have
Then L(s), the Lindelöf sum (Volkov 2001), is the limit of ƒ(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 MittagLeffler star.
If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence, then L(G(z)) = g(z) in the MittagLeffler star. Moreover, convergence to g(z) is uniform on compact subsets of the star.
Suppose that dμ is a measure on the nonnegative real line such that all the moments
are finite. If a_{0}+a_{1}+... is a series such that
converges for all nonnegative x, then the (dμ) sum of the series is defined to be the value of the integral
if it is defined. (Note that if the numbers μ_{n} increase too rapidly then they do not uniquely determine the measure μ.)
For example, if dμ = e^{–x}dx then μ_{n} = n!, and this gives one version of Borel summation.
If the series
converges for large real s and can be analytically continued along the real line to s=–1, then its value at s=–1 is called the zeta regularized sum of the series a_{1}+a_{2}+... In applications, the numbers a_{i} are sometimes the eigenvalues of a selfadjoint operator A with compact resolvant, and f(s) is then the trace of A^{–s}. For example, if A has eigenvalues 1, 2, 3, ... then f(s) is the Riemann zeta function, whose value at s=–1 is –1/12.
