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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 limit. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The limit of a sequence is one of the oldest concepts in Mathematical analysis.

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 counter example is the harmonic series

1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots =\sum_{n=1}^\infty\frac{1}{n}.

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 In Mathematics, the harmonic series is the Infinite series See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series Nicole Oresme, also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme (c

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. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function For example, Cesàro summation assigns Grandi's divergent series

1 - 1 + 1 - 1 + \cdots

the value 1/2. In Mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. The Infinite series 1 − 1 + 1 − 1 + &hellip or \sum_{n=0}^{\infin} (-1^nis sometimes called Grandi's series, after Italian Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided Other methods involve analytic continuations of related series. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, especially Quantum field theory, regularization is a method of dealing with infinite divergent and non-sensical expressions by introducing an auxiliary

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Theorems on methods for summing divergent series

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. In Mathematics, abelian and tauberian theorems relate to the meaningful assignment of a value as the "sum" of a class of Divergent series. In Mathematics, Abel's theorem for Power series relates a limit of a power series to the sum of its Coefficients It is named after Norwegian More interesting and in general more subtle are partial converse results, called tauberian theorems, from a prototype proved by Alfred Tauber. In Mathematics, abelian and tauberian theorems relate to the meaningful assignment of a value as the "sum" of a class of Divergent series. Alfred Tauber ( November 5, 1866 – circa 1942 was a mathematician who was born in Bratislava, and died in Theresienstadt concentration camp 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 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. In Mathematics, the Hahn–Banach theorem is a central tool in Functional analysis. 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. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which 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. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums In Mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. In Mathematics, a Borel summation is a generalisation of the usual notion of summation of a series The advent of Wiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebra methods in Fourier analysis. In Mathematics, Wiener's tauberian theorem is a 1932 result of Norbert Wiener. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series

Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. In Mathematics, extrapolation is the process of constructing new data points outside a Discrete set of known data points In Mathematics, a sequence transformation is an Operator acting on a given space of Sequences Sequence transformations include linear mappings such as Examples for such techniques are Padé approximants, Levin-type sequence transformations, and order-dependent mappings related to renormalization techniques for large-order perturbation theory in quantum mechanics. Padé approximant is the "best" approximation of a function by a Rational function of given order In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection This article describes perturbation theory as a general mathematical method Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

Properties of summation methods

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. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of So in evaluating a = a0 + a1 + a2 + . . . , we work with the sequence s, where s0 = a0 and sn+1 = sn + an. 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 series-summation method AΣ 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.

  1. Regularity. A summation method is regular if, whenever the sequence s converges to x, A(s) = x. Equivalently, the corresponding series-summation method evaluates AΣ(a) = x.
  2. Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that A(r + s) = A(r) + A(s) and A(ks) = k A(s), for k a scalar (real or complex. ) Since the terms an = sn+1−sn of the series a are linear functionals on the sequence s and vice versa, this is equivalent to AΣ being a linear functional on the terms of the series.
  3. Stability. If s is a sequence starting from s0 and s′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that s′n = sn+1 - s0, then A(s) is defined if and only if A(s′) is defined, and A(s) = s0 + A(s′). Equivalently, whenever a′n = an+1 for all n, then AΣ(a) = a0 + AΣ(a′).

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, as well as the order-dependent mappings of perturbative series based on renormalization techniques. In Mathematics, a sequence transformation is an Operator acting on a given space of Sequences Sequence transformations include linear mappings such as Padé approximant is the "best" approximation of a function by a Rational function of given order In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection

Axiomatic methods

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

\begin{align} G(r,c) & = \sum_{k=0}^\infty cr^k & & \\ & = c + \sum_{k=0}^\infty cr^{k+1} & & \mbox{ (stability) } \\ & = c + r \sum_{k=0}^\infty cr^k & & \mbox{ (linearity) } \\ & = c + r \, G(r,c), & & \mbox{ whence } \\ G(r,c) & = \frac{c}{1-r} , & & \\ \end{align}

can be evaluated regardless of convergence. In Mathematics, an Infinite geometric series of the form \sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots is divergent if and only 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 ∞.

Nörlund means

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

\frac{p_n}{p_0+p_1 + \cdots + p_n} \rightarrow 0.

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

t_m = \frac{p_m s_0 + p_{m-1}s_1 + \cdots + p_0 s_m}{p_0+p_1+\cdots+p_m}

then the limit of tn as n goes to infinity is an average called the Nörlund mean Np(s). Niels Erik Nörlund ( 26 October 1885 &ndash 4 July 1981) was a Danish mathematician

The Nörlund mean is regular, linear, and stable. Moreover, any two Nörlund means are consistent. The most significant of the Nörlund means are the Cesàro sums. Here, if we define the sequence pk by

p_n^k = {n+k-1 \choose k-1} = \frac{\Gamma(n+k)}{\Gamma(k)}

then the Cesàro sum Ck is defined by Ck(s) = N(pk)(s). Cesàro sums are Nörlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C0 is ordinary summation, and C1 is ordinary Cesàro summation. In Mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. Cesàro sums have the property that if h > k, then Ch is stronger than Ck.

Abelian means

Suppose λ = {λ0, λ1, λ2, …} is a strictly increasing sequence tending towards infinity, and that λ0 ≥ 0. Recall that an = sn+1 − sn is the associated series whose partial sums form the sequence s. Suppose

f(x) = \sum_{n=0}^\infty a_n \exp(-\lambda_n x)

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

A_\lambda(s) = \lim_{x \rightarrow 0^{+}} f(x).

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

Abelian means are regular, linear, and stable, but not always consistent between different choices of λ. However, some special cases are very important summation methods.

Abel summation

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

f(x) = \sum_{n=0}^\infty a_n \exp(-nx) = \sum_{n=0}^\infty a_n z^n,

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

A(s) = \lim_{z \rightarrow 1^{-}} \sum_{n=0}^\infty a_n z^n.

Abel summation is interesting in part because it is consistent with but more powerful than Cesàro summation: A(s) = Ck(s) whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.

Lindelöf summation

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

f(x) = a_1 + a_2 2^{-2x} + a_3 3^{-3x} + \cdots .

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. In Complex analysis, a branch of Mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the Complex plane obtained

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 Mittag-Leffler star. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives 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.

See also

References

Dictionary

divergent series

-noun

  1. (mathematics) An infinite series whose partial sums are divergent
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