Riemann series theorem

Riemann series theorem

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

 so that the series converges to any given value, or even diverges
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....

.

Definitions


A series converges if there exists a value such that the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in 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
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, and that is conditionally convergent. Let be a real number. Then there exists a permutation of the sequence 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.

Changing the sum


The alternating harmonic series is a classic example of a conditionally convergent series:


is convergent, while


is the ordinary harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

, which diverges. Although in standard presentation the alternating harmonic series converges to ln(2), 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 the terms:


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, the sum is composed of blocks of three:


This is indeed a rearrangement of the alternating harmonic series: every odd integer occurs once positively, and the even integers occur once each, negatively (half of them as multiples of 4, the other half as twice odd integers). Since


this series can in fact be written:


which is half the usual sum.

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
Euler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

, and where the notation o(1)
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

 denotes a quantity that depends upon the current variable (here, the variable is n) 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 (the alternating series itself corresponds to , the example in the preceding section corresponds to a = 1, b = 2):


Then the partial sum of order (a+b)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


For simplicity, this proof assumes first that an ≠ 0 for every n. The general case requires a simple modification, given below. 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 M be a positive real number. Take, in order, just enough positive terms so that their sum exceeds M. Suppose we require p terms – then the following statement is true:


This is possible for any M > 0 because the partial sums of tend to . Discarding the zero terms one may write


Now we add just enough negative terms , say q of them, so that the resulting sum is less than M. This is always possible because the partial sums of tend to . Now we have:


Again, one may write


with


Note that σ is injective, and that 1 belongs to the range of σ, either as image of 1 (if a1 > 0), or as image of (if a1 < 0). Now repeat the process of adding just enough positive terms to exceed M, starting with , and then adding just enough negative terms to be less than M, starting with . Extend σ in an injective manner, in order to cover all terms selected so far, and observe that must have been selected now or before, thus 2 belongs to the range of this extension. The process will have infinitely many such "changes of direction". One eventually obtains a rearrangement  . After the first change of direction, each partial sum of  differs from M by at most the absolute value or of the term that appeared at the latest change of direction. But converges, so as n tends to infinity, each of an, and go to 0. Thus, the partial sums of  tend to M, so the following is true:


The same method can be used to show convergence to M negative or zero.

One can now give a formal inductive definition of the rearrangement σ, that works in general. For every integer k ≥ 0, a finite set Ak of integers and a real number Sk are defined. For every k > 0, the induction defines the value σ(k), the set Ak consists of the values σ(j) for j ≤ k and Sk is the partial sum of the rearranged series. The definition is as follows:
  • For k = 0, the induction starts with A0 empty and S0 = 0.

  • For every k ≥ 0, there are two cases: if Sk ≤ M, then σ(k+1) is the smallest integer n ≥ 1 such that n is not in Ak and an ≥ 0; if Sk > M, then σ(k+1) is the smallest integer n ≥ 1 such that n is not in Ak and an < 0. In both cases one sets



It can be proved, using the reasonings above, that σ is a permutation of the integers and that the permuted series converges to the given real number M.

Generalization



Given a converging series of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, several cases can occur when considering the set of possible sums for all series obtained by rearranging (permuting) 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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 E, the set of sums of converging rearranged series is an affine subspace
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

of E.