In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, 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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
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 bic ...
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
converges if there exists a value
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 called ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s 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
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
is conditionally convergent. Let
be a real number. Then there exists a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
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
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:
is convergent, whereas
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,
:
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'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
:
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
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,
and
by:
:
That is, the series
includes all ''a''
''n'' positive, with all negative terms replaced by zeroes, and the series
includes all ''a''
''n'' 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
:
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
. 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 ''a''
''n'',
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 ''A''
''k'' of integers and a real number ''S''
''k'' are defined. For every ''k'' > 0, the induction defines the value
, the set ''A''
''k'' consists of the values
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
is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''
''k'' and ''a''
''n'' ≥ 0; if ''S''
''k'' > ''M'', then
is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''
''k'' and ''a''
''n'' < 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''.
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
(a similar argument can be used to show that
can also be attained).
Let
be the sequence of indexes such that each
is positive, and define
to be the indexes such that each
is negative (again assuming that
is never 0). Each natural number will appear in exactly one of the sequences
and
Let
be the smallest natural number such that
:
Such a value must exist since
the subsequence of positive terms of
diverges. Similarly, let
be the smallest natural number such that:
:
and so on. This leads to the permutation
:
And the rearranged series,
then diverges to
.
From the way the
were chosen, it follows that the sum of the first
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
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
Existence of a rearrangement that fails to approach any limit, finite or infinite
In fact, if
is conditionally convergent, then there is a rearrangement of it such that the partial sums of the rearranged series form a dense subset of
To construct such a rearrangement, first enumerate all rational numbers as
, then rearrange
such that the partial sum first approaches
until it is within distance
, then approaches
until it is within distance
, and so on.
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 let
With this notation, we have:
* If
is finite, then
. Here
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
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,