Riemann series theorem


In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent.
As an example, the series
converges to 0 ; but replacing all terms with their absolute values gives
which sums to infinity. Thus, the original series is conditionally convergent, and can be rearranged to give a series that converges to a different sum, such as
which evaluates to ln 2. More generally, using this procedure with p positives followed by q negatives gives the sum ln. Other rearrangements give other finite sums or do not converge to any sum.

History

It is a basic result that the sum of finitely many numbers does not depend on the order in which they are added. For example,. The observation that the sum of an infinite sequence of numbers can depend on the ordering of the summands is commonly attributed to Augustin-Louis Cauchy in 1833. He analyzed the alternating harmonic series, showing that certain rearrangements of its summands result in different limits. Around the same time, Peter Gustav Lejeune Dirichlet highlighted that such phenomena are ruled out in the context of absolute convergence, and gave further examples of Cauchy's phenomenon for some other series which fail to be absolutely convergent.
In the course of his analysis of Fourier series and the theory of Riemann integration, Bernhard Riemann gave a full characterization of the rearrangement phenomena. He proved that in the case of a convergent series which does not converge absolutely, rearrangements can be found so that the new series converges to any arbitrarily prescribed real number. Riemann's theorem is now considered as a basic part of the field of mathematical analysis.
For any series, one may consider the set of all possible sums, corresponding to all possible rearrangements of the summands. Riemann's theorem can be formulated as saying that, for a series of real numbers, this set is either empty, a single point, or the entire real number line. In this formulation, Riemann's theorem was extended by Paul Lévy and Ernst Steinitz to series whose summands are complex numbers or, even more generally, elements of a finite-dimensional real vector space. They proved that the set of possible sums forms a real affine subspace. Extensions of the Lévy–Steinitz theorem to series in infinite-dimensional spaces have been considered by a number of authors.

Definitions

A series converges if there exists a value such that the sequence of the partial sums
converges to. That is, for any ε > 0, there exists an integer N such that if n ≥ N, then
A series converges conditionally if the series converges but the series diverges.
A permutation is simply a bijection from the set of positive integers to itself. This means that if is a permutation, then for any positive integer there exists exactly one positive integer such that In particular, if, then.

Statement of the theorem

Suppose that is a sequence of real numbers, and that is conditionally convergent. Let be a real number. Then there exists a permutation such that
There also exists a permutation such that
The sum can also be rearranged to diverge to or to fail to approach any limit, finite or infinite.

Alternating harmonic series

Changing the sum

The alternating harmonic series is a classic example of a conditionally convergent series:is convergent, whereasis the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to, its terms can be arranged to converge to any number, or even to diverge.
One instance of this is as follows. Begin with the series written in the usual order,
and rearrange and regroup the terms as
where the pattern is: the first two terms are 1 and −1/2, whose sum is 1/2. The next term is −1/4. The next two terms are 1/3 and −1/6, whose sum is 1/6. The next term is −1/8. The next two terms are 1/5 and −1/10, whose sum is 1/10. In general, since every odd integer occurs once positively and every even integers occur once negatively, the sum is composed of blocks of three, each of which can be simplified as
Hence, the above series can in fact be written as
which is half the sum originally, and can only equate to the original sequence if the value were zero. This series can be demonstrated to be greater than zero by the proof of Leibniz's theorem using that the second partial sum is half. Alternatively, the value of which it converges to, cannot be zero. Hence, the value of the sequence is shown to depend on the order in which series is computed.
It is true that the sequence
contains all elements in the sequence
However, since the summation is defined as and, the order of the terms can influence the limit.

Getting an arbitrary sum

An efficient way to recover and generalize the result of the previous section is to use the fact that
where γ is the Euler–Mascheroni constant, and where the notation o(1) denotes a quantity that depends upon the current variable in such a way that this quantity goes to 0 when the variable tends to infinity.
It follows that the sum of q even terms satisfies
and by taking the difference, one sees that the sum of p odd terms satisfies
Suppose that two positive integers a and b are given, and that a rearrangement of the alternating harmonic series is formed by taking, in order, a positive terms from the alternating harmonic series, followed by b negative terms, and repeating this pattern at infinity :
Then the partial sum of order n of this rearranged series contains positive odd terms and negative even terms, hence
It follows that the sum of this rearranged series is
Suppose now that, more generally, a rearranged series of the alternating harmonic series is organized in such a way that the ratio between the number of positive and negative terms in the partial sum of order n tends to a positive limit r. Then, the sum of such a rearrangement will be
and this explains that any real number x can be obtained as sum of a rearranged series of the alternating harmonic series: it suffices to form a rearrangement for which the limit r is equal.

Proof

Existence of a rearrangement that sums to any positive real ''M''

Riemann's description of the theorem and its proof reads in full:
This can be given more detail as follows. Recall that a conditionally convergent series of real terms has both infinitely many negative terms and infinitely many positive terms. First, define two quantities, and by:
That is, the series includes all an positive, with all negative terms replaced by zeroes, and the series includes all an negative, with all positive terms replaced by zeroes. Since is conditionally convergent, both the 'positive' and the 'negative' series diverge. Let be any real number. Take just enough of the positive terms so that their sum exceeds. That is, let be the smallest positive integer such that
This is possible because the partial sums of the series tend to. Now let be the smallest positive integer such that
This number exists because the partial sums of tend to. Now continue inductively, defining as the smallest integer larger than such that
and so on. The result may be viewed as a new sequence
Furthermore, the partial sums of this new sequence converge to. This can be seen from the fact that for any,
with the first inequality holding due to the fact that has been defined as the smallest number larger than which makes the second inequality true; as a consequence, it holds that
Since the right-hand side converges to zero due to the assumption of conditional convergence, this shows that the 'th partial sum of the new sequence converges to as increases. Similarly, the 'th partial sum also converges to. Since the 'th, 'th,... 'th partial sums are valued between the 'th and 'th partial sums, it follows that the whole sequence of partial sums converges to.
Every entry in the original sequence appears in this new sequence whose partial sums converge to. Those entries of the original sequence which are zero will appear twice in the new sequence, and every second such appearance can be removed, which does not affect the summation in any way. The new sequence is thus a permutation of the original sequence.

Existence of a rearrangement that diverges to infinity

Let be a conditionally convergent series. The following is a proof that there exists a rearrangement of this series that tends to .
The above proof of Riemann's original formulation only needs to be modified so that is selected as the smallest integer larger than such that
and with selected as the smallest integer larger than such that
The choice of on the left-hand sides is immaterial, as it could be replaced by any sequence increasing to infinity. Since converges to zero as increases, for sufficiently large there is
and this proves that the sequence of partial sums of the new sequence diverge to infinity.

Existence of a rearrangement that fails to approach any limit, finite or infinite

The above proof only needs to be modified so that is selected as the smallest integer larger than such that
and with selected as the smallest integer larger than such that
This directly shows that the sequence of partial sums contains infinitely many entries which are larger than 1, and also infinitely many entries which are less than, so that the sequence of partial sums cannot converge.

Generalizations

Sierpiński theorem

Given an infinite series, we may consider a set of "fixed points", and study the real numbers that the series can sum to if we are only allowed to permute indices in. That is, we letWith this notation, we have:
  • If is finite, then. Here means symmetric difference.
  • If then.
  • If the series is an absolutely convergent sum, then for any.
  • If the series is a conditionally convergent sum, then by Riemann series theorem,.
Sierpiński proved that rearranging only the positive terms one can obtain a series converging to any prescribed value less than or equal to the sum of the original series, but larger values in general can not be attained. That is, let be a conditionally convergent sum, then contains, but there is no guarantee that it contains any other number.
More generally, let be an ideal of, then we can define.
Let be the set of all asymptotic density zero sets, that is,. It's clear that is an ideal of.
Proof sketch: Given, a conditionally convergent sum, construct some such that and are both conditionally convergent. Then, rearranging suffices to converge to any number in.
Filipów and Szuca proved that other ideals also have this property.

Steinitz's theorem

Given a converging series of complex numbers, several cases can occur when considering the set of possible sums for all series obtained by rearranging the terms of that series:
  • the series may converge unconditionally; then, all rearranged series converge, and have the same sum: the set of sums of the rearranged series reduces to one point;
  • the series may fail to converge unconditionally; if S denotes the set of sums of those rearranged series that converge, then, either the set S is a line L in the complex plane C, of the form or the set S is the whole complex plane C.
More generally, given a converging series of vectors in a finite-dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E.