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Encyclopedia > Divergent series

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. 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. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ... For other senses of this word, see sequence (disambiguation). ... In mathematics, a series is a sum of a sequence of terms. ... 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 See harmonic series (music) for the (related) musical concept. ...

1 + frac{1}{2} + frac{1}{3} + frac{1}{4} + frac{1}{5} + cdots =sum_{n=1}^inftyfrac{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. ... Nicolas 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. For example, Cesàro summation assigns Grandi's divergent series In mathematics, a partial function is a relation that associates each element of a set (sometimes called its domain) with at most one element of another (possibly the same) set, called the codomain. ... 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 + · · · or is sometimes called Grandis series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. ...

1 - 1 + 1 - 1 + cdots

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. 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 by the number of items in the list. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... 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). ...

Contents

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. 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 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). 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 has its beginnings in the rigorous formulation of calculus. ... In mathematics, a divergent series is an infinite series that does not converge. ... 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. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ...


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

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 = 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. In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ...

  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+1sn 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 sn = sn+1 - s0, then A(s) is defined if and only if A(s′) is defined, and A(s) = s0 + A(s′). Equivalently, whenever an = 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. 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. ...


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 In mathematics, an infinite geometric series of the form is divergent if and only if | r | â‰¥ 1. ...

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. 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 positive real number, 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).


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. Cesàro sums have the property that if h > k, then Ch is stronger than Ck. 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 = {λ0, λ1, λ2, …} is a strictly increasing sequence tending towards infinity, and that λ0 ≥ 0. Recall that an = sn+1sn 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 to superficially divergent sums. ...


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^{-}} 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. Illustration of the Mittag-Leffler star (the region bounded by the blue contour). ...


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. 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. ...


See also

  • 1 − 2 + 3 − 4 + · · ·
  • 1 − 2 + 4 − 8 + · · ·
  • 1 + 1 + 1 + 1 + · · ·
  • 1 + 2 + 3 + 4 + · · ·
  • 1 + 2 + 4 + 8 + · · ·

The first few thousand terms and partial sums of 1 − 2 + 3 − 4 + … In mathematics, 1 − 2 + 3 − 4 + … is the infinite series whose terms are the successive positive integers, given alternating signs. ... In mathematics, 1 − 2 + 4 − 8 + … is the infinite series whose terms are the successive powers of two with alternating signs. ... 1 + 1 + 1 + 1 + · · ·, also written , is a divergent series. ... 1 + 2fuckyouadlkfjl;asdjf;dajslfjadlsfjjj100000000000009999999999999999999999999999999999999· · ·, also written , is a divergent series. ... In mathematics, 1 + 2 + 4 + 8 + · · · is the infinite series whose terms are the successive powers of two. ...

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.

  Results from FactBites:
 
Divergent series - Wikipedia, the free encyclopedia (371 words)
That is, divergent series and convergent series are antonyms.
The divergence of the harmonic series was proved by the medieval mathematician Nicole Oresme.
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.
Infinite series - definition of Infinite series in Encyclopedia (1539 words)
In mathematics, a series is a sum of a sequence of terms.
Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
  More results at FactBites »

 
 

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