Riemann rearrangement theorem
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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 In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of real numbers is
conditionally convergent In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Definition More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if \lim_\,\s ...
, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This implies that a series of real numbers is
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
it is unconditionally convergent. As an example, the series 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ⋯ converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ⋯, which sums to infinity. Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum: 1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ⋯ = ln 2. More generally, using this procedure with ''p'' positives followed by ''q'' negatives gives the sum ln(''p''/''q''). Other rearrangements give other finite sums or do not converge to any sum.


Definitions

A series \sum_^\infty a_n converges if there exists a value \ell such that the
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of the partial sums :(S_1, S_2, S_3, \ldots), \quad S_n = \sum_^n a_k, converges to \ell. That is, for any ''ε'' > 0, there exists an integer ''N'' such that if ''n'' ≥ ''N'', then :\left\vert S_n - \ell \right\vert \le \epsilon. A series converges conditionally if the series \sum_^\infty a_n converges but the series \sum_^\infty \left\vert a_n \right\vert diverges. A permutation is simply a bijection from the set of positive integers to itself. This means that if \sigma is a permutation, then for any positive integer b, there exists exactly one positive integer a such that \sigma (a) = b. In particular, if x \ne y, then \sigma (x) \ne \sigma (y).


Statement of the theorem

Suppose that (a_1, a_2, a_3, \ldots) is a sequence of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, and that \sum_^\infty a_n is conditionally convergent. Let M be a real number. Then there exists a permutation \sigma such that :\sum_^\infty a_ = M. There also exists a permutation \sigma such that :\sum_^\infty a_ = \infty. The sum can also be rearranged to diverge to -\infty or to fail to approach any limit, finite or infinite.


Alternating harmonic series


Changing the sum

The
alternating harmonic series In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, wher ...
is a classic example of a conditionally convergent series: \sum_^\infty \frac is convergent, whereas \sum_^\infty \left, \frac \ = \sum_^\infty \frac is 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, :1 - \frac + \frac - \frac + \cdots and rearrange the terms: :1 - \frac - \frac + \frac - \frac - \frac + \frac - \frac - \frac + \cdots 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: :\frac - \frac - \frac,\quad k = 1, 2, \dots. 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 :\frac - \frac = \frac, this series can in fact be written: :\begin &\frac - \frac + \frac - \frac + \frac + \cdots + \frac - \frac + \cdots \\ =& \frac\left(1 - \frac + \frac - \cdots\right) = \frac \ln(2) \end 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 :1 + + + \cdots + = \gamma + \ln n + o(1), where ''γ'' is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
, and where the notation ''o''(1) 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 : + + + \cdots + = \, \gamma + \ln q + o(1), and by taking the difference, one sees that the sum of ''p'' odd terms satisfies : + + + \cdots + = \, \gamma + \ln p + \ln 2 + o(1). 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): : + + \cdots + - - - \cdots - + + \cdots + - - \cdots Then the partial sum of order (''a''+''b'')''n'' of this rearranged series contains positive odd terms and negative even terms, hence :S_ = \ln p + \ln 2 - \ln q + o(1) = \ln(a/b) + \ln 2 + o(1). It follows that the sum of this rearranged series is : \ln(a/b) + \ln 2 = \ln\left( 2 \sqrt \right). 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 :\ln\left( 2 \sqrt \right), 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''

For simplicity, this proof assumes first that ''a''''n'' ≠ 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, a_n^+ and a_n^- by: :a_^ = \frac, \quad a_^ = \frac. That is, the series \sum_^\infty a_n^ includes all ''a''''n'' positive, with all negative terms replaced by zeroes, and the series \sum_^\infty a_n^ includes all ''a''''n'' negative, with all positive terms replaced by zeroes. Since \sum_^\infty a_n 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 a_^ so that their sum exceeds ''M''. Suppose we require ''p'' terms – then the following statement is true: :\sum_^ a_^ \leq M < \sum_^ a_^. This is possible for any ''M'' > 0 because the partial sums of a_^ tend to +\infty. Discarding the zero terms one may write :\sum_^ a_^ = a_ + \cdots + a_, \quad a_ > 0, \ \ \sigma(1) < \dots < \sigma(m_1) = p. Now we add just enough negative terms a_^, say ''q'' of them, so that the resulting sum is less than ''M''. This is always possible because the partial sums of a_^ tend to -\infty. Now we have: :\sum_^ a_^ + \sum_^ a_^ < M \leq \sum_^ a_^ + \sum_^ a_^. Again, one may write :\sum_^ a_^ + \sum_^ a_^ = a_ + \cdots + a_ + a_ + \cdots + a_, with : \sigma(m_1+1) < \dots < \sigma(n_1) = q. The map ''σ'' is injective, and 1 belongs to the range of ''σ'', either as image of 1 (if ''a''1 > 0), or as image of (if ''a''1 < 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 \sum. After the first change of direction, each partial sum of \sum differs from ''M'' by at most the absolute value a_^ or , a_^, of the term that appeared at the latest change of direction. But \sum converges, so as ''n'' tends to infinity, each of ''a''''n'', a_^ and a_^ go to 0. Thus, the partial sums of \sum tend to ''M'', so the following is true: :\sum_^\infty a_ = M. 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 ''A''''k'' of integers and a real number ''S''''k'' are defined. For every ''k'' > 0, the induction defines the value \sigma(k), the set ''A''''k'' consists of the values \sigma(j) for ''j'' ≤ ''k'' and ''S''''k'' is the partial sum of the rearranged series. The definition is as follows: * For ''k'' = 0, the induction starts with ''A''0 empty and ''S''0 = 0. * For every ''k'' ≥ 0, there are two cases: if ''S''''k'' ≤ ''M'', then \sigma(k+1) is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''''k'' and ''a''''n'' ≥ 0; if ''S''''k'' > ''M'', then \sigma(k+1) is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''''k'' and ''a''''n'' < 0. In both cases one sets A_ = A_k \cup \ \, ; \quad S_ = S_k + a_. 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''.


Existence of a rearrangement that diverges to infinity

Let \sum_^\infty a_i be a conditionally convergent series. The following is a proof that there exists a rearrangement of this series that tends to \infty (a similar argument can be used to show that -\infty can also be attained). Let p_1 < p_2 < p_3 < \cdots be the sequence of indexes such that each a_ is positive, and define n_1 < n_2 < n_3 < \cdots to be the indexes such that each a_ is negative (again assuming that a_i is never 0). Each natural number will appear in exactly one of the sequences (p_i) and (n_i). Let b_1 be the smallest natural number such that :\sum_^ a_ \geq , a_, + 1. Such a value must exist since (a_), the subsequence of positive terms of (a_i), diverges. Similarly, let b_2 be the smallest natural number such that: :\sum_^ a_ \geq , a_, + 1, and so on. This leads to the permutation :(\sigma(1),\sigma(2),\sigma(3),\ldots) = (p_1, p_2, \ldots, p_, n_1, p_, p_, \ldots, p_, n_2, \ldots). And the rearranged series, \sum_^\infty a_, then diverges to \infty. From the way the b_i were chosen, it follows that the sum of the first b_1+1 terms of the rearranged series is at least 1 and that no partial sum in this group is less than 0. Likewise, the sum of the next b_2 - b_1 + 1 terms is also at least 1, and no partial sum in this group is less than 0 either. Continuing, this suffices to prove that this rearranged sum does indeed tend to \infty.


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

In fact, if \sum _^ a_ is conditionally convergent, then there is a rearrangement of it such that the partial sums of the rearranged series form a dense subset of \Reals. To construct such a rearrangement, first enumerate all rational numbers as r_1, r_2, ..., then rearrange \sum _^ a_ such that the partial sum first approaches r_1 until it is within distance 1/1, then approaches r_2 until it is within distance 1/2, and so on.


Generalizations


Sierpiński theorem

Given an infinite series a = (a_1, a_2, ...), we may consider a set of "fixed points" I \subset \N, and study the real numbers that the series can sum to if we are only allowed to permute indices in I. That is, we letS(a, I) = \left\With this notation, we have: * If I \Delta I' is finite, then S(a, I) = S(a, I'). Here \Delta means
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
. * If I \subset I' then S(a, I) \subset S(a, I'). * If the series is an absolutely convergent sum, then S(a, I) = \left\ for any I. * If the series is a conditionally convergent sum, then by Riemann series theorem, S(a, \N) = \infty, +\infty/math>. 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 a be a conditionally convergent sum, then S(a, \) contains \left \infty, \sum_ a_n\right/math>, but there is no guarantee that it contains any other number. More generally, let J be an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of \N, then we can define S(a, J) = \cup_ S(a, I). Let J_d be the set of all asymptotic density zero sets I\subset \N, that is, \lim_\frac = 0. It's clear that J_d is an ideal of \N. Proof sketch: Given a, a conditionally convergent sum, construct some I\in J_d such that \sum_a_n and \sum_a_n are both conditionally convergent. Then, rearranging \sum_a_n suffices to converge to any number in \infty, +\infty/math>. Filipów and Szuca proved that other ideals also have this property.


Steinitz's theorem

Given a converging series \sumof
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, several cases can occur when considering the set of possible sums for all series \sum obtained by rearranging (permuting) the terms of that series: * the series \sum 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 \summay 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 L = \, \quad a, b \in \Complex, \ b \ne 0, 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''E'', the set of sums of converging rearranged series is an
affine subspace In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of ''E''.


See also

*


References

*Apostol, Tom (1975). ''Calculus, Volume 1: One-variable Calculus, with an Introduction to Linear Algebra.'' * * * *Weisstein, Eric (2005)
Riemann Series Theorem
Retrieved May 16, 2005. {{Bernhard Riemann Mathematical series Theorems in analysis Permutations Summability theory Bernhard Riemann