Divergent series

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In mathematics, a divergent series is an infinite series that does not converge.

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

1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 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.

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

1 - 1 + 1 - 1 + \cdots

the value ½. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

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[edit] 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).

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 and Levin-type sequence transformations.

[edit] 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.

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.

[edit] Nõrlund 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 the associated Nõrlund mean Np(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

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) = Npk(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 Ch is stronger than Ck.

[edit] Abelian means

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

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

[edit] 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; if Ck(s) = a for any positive k, then A(s) = a. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.

[edit] 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.

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.

[edit] See also

[edit] 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.
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