In
mathematics, 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 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
series
is said to converge absolutely if
for some real number
Similarly, an
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
,
is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called
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 ...
, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: 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 ...
converges to
while its rearrangement
(in which the repeating pattern of signs is two positive terms followed by one negative term) converges to
Background
In finite sums, the order in which terms are added does not matter. 1 + 2 + 3 is the same as 3 + 2 + 1. However, this is not true when adding infinitely many numbers, and wrongly assuming that it is true can lead to apparent paradoxes. One classic example is the alternating sum
whose terms alternate between +1 and -1. What is the value of S? One way to evaluate S is to group the first and second term, the third and fourth, and so on:
But another way to evaluate S is to leave the first term alone and group the second and third term, then the fourth and fifth term, and so on:
This leads to an apparent paradox: does
or
?
The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means
and
are not equal. In fact, the series
does not
converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
, so S does not have a value to find in the first place. A series that is absolutely convergent does not have this problem: rearranging its terms does not change the value of the sum.
Definition for real and complex numbers
A sum of real numbers or complex numbers
is absolutely convergent if the sum of the absolute values of the terms
converges.
Sums of more general elements
The same definition can be used for series
whose terms
are not numbers but rather elements of an arbitrary
abelian topological group. In that case, instead of using the
absolute value, the definition requires the group to have a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, which is a positive real-valued function
on an abelian group
(written
additively, with identity element 0) such that:
# The norm of the identity element of
is zero:
# For every
implies
# For every
# For every
In this case, the function
induces the structure of a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
(a type of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) on
Then, a
-valued series is absolutely convergent if
In particular, these statements apply using the norm
(
absolute value) in the space of real numbers or complex numbers.
In topological vector spaces
If
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) and
is a (possibly
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
) family in
then this family is absolutely summable if
#
is summable in
(that is, if the limit
of the
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
converges in
where
is the
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
of all finite subsets of
directed by inclusion
and
), and
# for every continuous seminorm
on
the family
is summable in
If
is a normable space and if
is an absolutely summable family in
then necessarily all but a countable collection of
's are 0.
Absolutely summable families play an important role in the theory of
nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
s.
Relation to convergence
If
is
complete with respect to the metric
then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality.
In particular, for series with values in any
Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space.
If a series is convergent but not absolutely convergent, it is called
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 ...
. An example of a conditionally convergent series is 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 ...
. Many standard tests for divergence and convergence, most notably including the
ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
and the
root test
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity
:\limsup_\sqrt
where a_n are the terms of the series, and states that the series converges absolutely if ...
, demonstrate absolute convergence. This is because a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
is absolutely convergent on the interior of its disk of convergence.
Proof that any absolutely convergent series of complex numbers is convergent
Suppose that
is convergent. Then equivalently,
is convergent, which implies that
and
converge by termwise comparison of non-negative terms. It suffices to show that the convergence of these series implies the convergence of
and
for then, the convergence of
would follow, by the definition of the convergence of complex-valued series.
The preceding discussion shows that we need only prove that convergence of
implies the convergence of
Let
be convergent. Since
we have
Since
is convergent,
is a
bounded monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
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 partial sums, and
must also converge. Noting that
is the difference of convergent series, we conclude that it too is a convergent series, as desired.
Alternative proof using the Cauchy criterion and triangle inequality
By applying the Cauchy criterion for the convergence of a complex series, we can also prove this fact as a simple implication of the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
. By the
Cauchy criterion,
converges if and only if for any
there exists
such that
for any
But the triangle inequality implies that
so that
for any
which is exactly the Cauchy criterion for
Proof that any absolutely convergent series in a Banach space is convergent
The above result can be easily generalized to every
Banach space Let
be an absolutely convergent series in
As
is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of real numbers, for any
and large enough
natural number
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
it holds:
By the triangle inequality for the norm , one immediately gets:
which means that
is a Cauchy sequence in
hence the series is convergent in
Rearrangements and unconditional convergence
Real and complex numbers
When a series of real or complex numbers is absolutely convergent, any rearrangement or reordering of that series' terms will still converge to the same value. This fact is one reason absolutely convergent series are useful: showing a series is absolutely convergent allows terms to be paired or rearranged in convenient ways without changing the sum's value.
The
Riemann rearrangement 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 ...
shows that the converse is also true: every real or complex-valued series whose terms cannot be reordered to give a different value is absolutely convergent.
Series with coefficients in more general space
The term
unconditional convergence In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not a ...
is used to refer to a series where any rearrangement of its terms still converges to the same value. For any series with values in a normed abelian group
, as long as
is complete, every series which converges absolutely also converges unconditionally.
Stated more formally:
For series with more general coefficients, the converse is more complicated. As stated in the previous section, for real-valued and complex-valued series, unconditional convergence always implies absolute convergence. However, in the more general case of a series with values in any normed abelian group
, the converse does not always hold: there can exist series which are not absolutely convergent, yet unconditionally convergent.
For example, in the
Banach space ℓ
∞, one series which is unconditionally convergent but not absolutely convergent is:
where
is an orthonormal basis. A theorem of
A. Dvoretzky and
C. A. Rogers asserts that every infinite-dimensional Banach space has an unconditionally convergent series that is not absolutely convergent.
Proof of the theorem
For any
we can choose some
such that:
Let
where
so that
is the smallest natural number such that the list
includes all of the terms
(and possibly others).
Finally for any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
let
so that
and thus
This shows that
that is:
Q.E.D.
Products of series
The
Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infini ...
of two series converges to the product of the sums if at least one of the series converges absolutely. That is, suppose that
The Cauchy product is defined as the sum of terms
where:
If the
or
sum converges absolutely then
Absolute convergence over sets
A generalization of the absolute convergence of a series, is the absolute convergence of a sum of a function over a set. We can first consider a countable set
and a function
We will give a definition below of the sum of
over
written as
First note that because no particular enumeration (or "indexing") of
has yet been specified, the series
cannot be understood by the more basic definition of a series. In fact, for certain examples of
and
the sum of
over
may not be defined at all, since some indexing may produce a conditionally convergent series.
Therefore we define
only in the case where there exists some bijection
such that
is absolutely convergent. Note that here, "absolutely convergent" uses the more basic definition, applied to an indexed series. In this case, the value of the sum of
over
is defined by
Note that because the series is absolutely convergent, then every rearrangement is identical to a different choice of bijection
Since all of these sums have the same value, then the sum of
over
is well-defined.
Even more generally we may define the sum of
over
when
is uncountable. But first we define what it means for the sum to be convergent.
Let
be any set, countable or uncountable, and
a function. We say that the sum of
over
converges absolutely if
There is a theorem which states that, if the sum of
over
is absolutely convergent, then
takes non-zero values on a set that is at most countable. Therefore, the following is a consistent definition of the sum of
over
when the sum is absolutely convergent.
Note that the final series uses the definition of a series over a countable set.
Some authors define an iterated sum
to be absolutely convergent if the iterated series
This is in fact equivalent to the absolute convergence of
That is to say, if the sum of
over
converges absolutely, as defined above, then the iterated sum
converges absolutely, and vice versa.
Absolute convergence of integrals
The
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of a real or complex-valued function is said to converge absolutely if
One also says that
is absolutely integrable. The issue of absolute integrability is intricate and depends on whether the
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 rig ...
,
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, or
Kurzweil-Henstock (gauge) integral is considered; for the Riemann integral, it also depends on whether we only consider integrability in its proper sense (
and
both
bounded), or permit the more general case of improper integrals.
As a standard property of the Riemann integral, when