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 ...
, 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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and its generalization,
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
. Series are used in most areas of mathematics, even for studying finite structures (such as in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
) through
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
s. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
and
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, ma ...
could produce a finite result was considered
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
ical. This paradox was resolved using the concept of a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
during the 17th century.
Zeno's paradox
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plur ...
of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.
Zeno
Zeno ( grc, Ζήνων) may refer to:
People
* Zeno (name), including a list of people and characters with the name
Philosophers
* Zeno of Elea (), philosopher, follower of Parmenides, known for his paradoxes
* Zeno of Citium (333 – 264 BC), ...
concluded that Achilles could ''never'' reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.
In modern terminology, any (ordered)
infinite 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 t ...
of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like
or, using the summation sign,
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if 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 ...
to which the terms and their finite sums belong has a notion of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as tends to
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
(if the limit exists) of the finite sums of the first terms of the series, which are called the th partial sums of the series. That is,
When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.
The notation denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by both the
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
—the process of adding—and its result—the ''sum'' of and .
Generally, the terms of a series come from a ring, often the field of the
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 or the field of the
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. In this case, the set of all series is itself a ring (and even an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
), in which the addition consists of adding the series term by term, and the multiplication is 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 ...
.
Basic properties
An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form
where is any ordered
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 ...
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
). This is an expression that is obtained from the list of terms by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
If an abelian group of terms has a concept of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
(e.g., if it is 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 setti ...
), then some series, the
convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted
:S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k.
The th partial ...
, can be interpreted as having a value in , called the ''sum of the series''. This includes the common cases from
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, in which the group is the field 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 or the field of
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. Given a series , its th partial sum is
By definition, the series ''converges'' to the limit (or simply ''sums'' to ), if the sequence of its partial sums has a limit . In this case, one usually writes
A series is said to be ''convergent'' if it converges to some limit, or ''divergent'' when it does not. The value of this limit, if it exists, is then the value of the series.
Convergent series
A series is said to converge or to ''be convergent'' when the sequence of partial sums has a finite
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
. If the limit of is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series
An easy way that an infinite series can converge is if all the are zero for sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example
It is possible to "visualize" its convergence on the
real number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
: we can imagine a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
. If the series is denoted , it can be seen that
Therefore,
The idiom can be extended to other, equivalent notions of series. For instance, a
recurring decimal
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if a ...
, as in
encodes the series
Since these series always converge to
real numbers
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 re ...
(because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the
arithmetic operations
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
'' is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example:
In general, the geometric series
converges
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 ...
, in which case it converges to with f(n)=a_ for all n, then \sum a_ converges if and only if the integral">,\infty) with f(n)=a_ for all n, then \sum a_ converges if and only if the integral \int_^ f(x) \, dx is finite.
* Cauchy's condensation test: If a_ is non-negative and non-increasing, then the two series \sum a_ and \sum 2^ a_ are of the same nature: both convergent, or both divergent.
* Alternating series test: A series of the form \sum (-1)^ a_ (with a_ > 0) is called ''alternating''. Such a series converges if 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 ...
''a_'' is monotone decreasing and converges to 0. The converse is in general not true.
* For some specific types of series there are more specialized convergence tests, for instance for
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
A series of real- or complex-valued functions
\sum_^\infty f_n(x)converges pointwise on a set ''E'', if the series converges for each ''x'' in ''E'' as an ordinary series of real or complex numbers. Equivalently, the partial sums
s_N(x) = \sum_^N f_n(x)
converge to ''ƒ''(''x'') as ''N'' → ∞ for each ''x'' ∈ ''E''.
A stronger notion of convergence of a series of functions is the
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
. A series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum,
, s_N(x) - f(x),
can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''.
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ''ƒ''''n'' are integrable on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test,
Dini's test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.
Definition
Le ...
, and the
Cauchy criterion
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analy ...
.
More sophisticated types of convergence of a series of functions can also be defined. In
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, for instance, a series of functions converges
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
if it converges pointwise except on a certain set of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
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 setti ...
structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set ''E'' to a limit function ''ƒ'' provided
\int_E \left, s_N(x)-f(x)\^2\,dx \to 0
as ''N'' → ∞.
Power series
:
A power series is a series of the form
\sum_^\infty a_n(x-c)^n.
The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
at a point ''c'' of a function is a power series that, in many cases, converges to the function in a neighborhood of ''c''. For example, the series
\sum_^ \frac
is the Taylor series of e^x at the origin and converges to it for every ''x''.
Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''c'' in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
, and can in principle be determined from the asymptotics of the coefficients ''a''''n''. The convergence is uniform on
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
Formal power series
While many uses of power series refer to their sums, it is also possible to treat power series as ''formal sums'', meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
to describe and study
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 ...
s that are otherwise difficult to handle, for example, using the method of
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
s.
Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
,
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
,
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, so that the formal power series can be added term-by-term and multiplied via 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 ...
. In this case the algebra of formal power series is the total algebra of the
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
of
natural numbers
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 ...
over the underlying term ring. If the underlying term ring is a
differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.
Laurent series
Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form
\sum_^\infty a_n x^n.
If such a series converges, then in general it does so in an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
Dirichlet series
:
A
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
is one of the form
\sum_^\infty ,
where ''s'' is a
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 ...
. For example, if all ''a''''n'' are equal to 1, then the Dirichlet series is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
\zeta(s) = \sum_^\infty \frac.
Like the zeta function, Dirichlet series in general play an important role in
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
. Generally a Dirichlet series converges if the real part of ''s'' is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
outside the domain of convergence by
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
. For example, the Dirichlet series for the zeta function converges absolutely when Re(''s'') > 1, but the zeta function can be extended to a holomorphic function defined on \Complex\setminus\ with a simple pole at 1.
This series can be directly generalized to
general Dirichlet series In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
: \sum_^\infty a_n e^,
where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends ...
.
Trigonometric series
A series of functions in which the terms are
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s is called a trigonometric series:
\frac12 A_0 + \sum_^\infty \left(A_n\cos nx + B_n \sin nx\right).
The most important example of a trigonometric series is the
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of a function.
History of the theory of infinite series
Development of infinite series
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
mathematician
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the
method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
to calculate the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
under the arc of a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
with the summation of an infinite series, and gave a remarkably accurate approximation of π.
Mathematicians from Kerala, India studied infinite series around 1350 CE.
In the 17th century, James Gregory worked in the new
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system on infinite series and published several
Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ...
. In 1715, a general method for constructing the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
for all functions for which they exist was provided by
Brook Taylor
Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis.
Life and work
Brook Taylor ...
.
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
q-series
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer s ...
.
Convergence criteria
The investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series
1 + \fracx + \fracx^2 + \cdots
on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by Gregory (1668).
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of
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 ...
by his expansion of a complex function in such a form.
Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
(1826) in his memoir on the binomial series1 + \fracx + \fracx^2 + \cdots
corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and
the same may be said of
Raabe The last name Raabe specifically originates from Prussia, derived from a Prussian warrior clans' symbol: a raven, which was one of the four beasts of war. During Prussia's decimation, most of these warriors intermarried with the Danish, and slowly m ...
(1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose
logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have
shown to fail within a certain region; of
Bertrand
Bertrand may refer to:
Places
* Bertrand, Missouri, US
* Bertrand, Nebraska, US
* Bertrand, New Brunswick, Canada
* Bertrand Township, Michigan, US
* Bertrand, Michigan
* Bertrand, Virginia, US
* Bertrand Creek, state of Washington
* Saint-Ber ...
(1842), Bonnet
(1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852),
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
(1852), and Arndt
(1853).
General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various
contributions to the theory of functions, Dini (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
Uniform convergence
The theory of
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it
successfully were Seidel and Stokes (1847–48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.
Semi-convergence
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not
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 ...
.
Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's functionF(x) = 1^n + 2^n + \cdots + (x - 1)^n.Genocchi (1852) has further contributed to the theory.
Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into
prominence.
Fourier series
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
(1702) and his brother
Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPoinsot, Schröter, Glaisher, and Kummer.
Fourier (1807) set for himself a different problem, to
expand a given function of ''x'' in terms of the sines or cosines of
multiples of ''x'', a problem which he embodied in his '' Théorie analytique de la chaleur'' (1822). Euler had already given the formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. Poisson (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for Cauchy (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see convergence of Fourier series). Dirichlet's treatment ('' Crelle'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine,
Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
, Schläfli, and
du Bois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were Dini,
Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Her ...
Asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
, otherwise
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
s, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.
Divergent series
Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
, (''C'',''k'') summation,
Abel summation
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 finite limit.
If a series converges, the individual terms of the series must ...
, and
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several va ...
, in increasing order of generality (and hence applicable to increasingly divergent series).
A variety of general results concerning possible summability methods are known. The
Silverman–Toeplitz theorem
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a con ...
characterizes ''matrix summability methods'', which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns
Banach limit
In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\in ...
s.
Summations over arbitrary index sets
Definitions may be given for sums over an arbitrary index set I. There are two main differences with the usual notion of series: first, there is no specific order given on the set I; second, this set I may be uncountable. The notion of convergence needs to be strengthened, because the concept of
conditional convergence 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_\,\ ...
depends on the ordering of the index set.
If a : I \mapsto G is a function from an
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
I to a set G, then the "series" associated to a is the formal sum of the elements a(x) \in G over the index elements x \in I denoted by the
\sum_ a(x).
When the index set is the natural numbers I=\N, the function a : \N \mapsto G is a
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 ...
denoted by a(n) = a_n. A series indexed on the natural numbers is an ordered formal sum and so we rewrite \sum_ as \sum_^ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
\sum_^ a_n = a_0 + a_1 + a_2 + \cdots.
Families of non-negative numbers
When summing a family \left\ of non-negative real numbers, define
\sum_a_i = \sup \left\ \in , +\infty
When the supremum is finite then the set of i \in I such that a_i > 0 is countable. Indeed, for every n \geq 1, the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
\left, A_n\ of the set A_n = \left\ is finite because
\frac \, \left, A_n\ = \sum_ \frac \leq \sum_ a_i \leq \sum_ a_i < \infty.
If I is countably infinite and enumerated as I = \left\ then the above defined sum satisfies
\sum_ a_i = \sum_^ a_,
provided the value \infty is allowed for the sum of the series.
Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.
Abelian topological groups
Let a : I \to X be a map, also denoted by \left(a_i\right)_, from some non-empty set I into a Hausdorff
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
X.
Let \operatorname(I) be the collection of all
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of I, with \operatorname(I) viewed as a
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 has ...
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two topo ...
.
The family \left(a_i\right)_, is said to be if the following
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, which is denoted by \sum_ a_i and is called the of \left(a_i\right)_, exists in X:\sum_ a_i := \lim_ \ \sum_ a_i = \lim \left\
Saying that the sum S := \sum_ a_i is the limit of finite partial sums means that for every neighborhood V of the origin in X, there exists a finite subset A_0 of I such that
S - \sum_ a_i \in V \qquad \text \; A \supseteq A_0.
Because \operatorname(I) is not
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
, this is not a
limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limi ...
of partial sums, but rather of a net.
For every neighborhood W of the origin in X, there is a smaller neighborhood V such that V - V \subseteq W. It follows that the finite partial sums of an unconditionally summable family \left(a_i\right)_, form a , that is, for every neighborhood W of the origin in X, there exists a finite subset A_0 of I such that
\sum_ a_i - \sum_ a_i \in W \qquad \text \; A_1, A_2 \supseteq A_0,
which implies that a_i \in W for every i \in I \setminus A_0 (by taking A_1 := A_0 \cup \ and A_2 := A_0).
When X is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, a family \left(a_i\right)_ is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and \left(a_i\right)_, is unconditionally summable in X, then for every subset J \subseteq I, the corresponding subfamily \left(a_j\right)_, is also unconditionally summable in X.
When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = \R.
If a family \left(a_i\right)_ in X is unconditionally summable then for every neighborhood W of the origin in X, there is a finite subset A_0 \subseteq I such that a_i \in W for every index i not in A_0. If X is a
first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
then it follows that the set of i \in I such that a_i \neq 0 is countable. This need not be true in a general abelian topological group (see examples below).
Unconditionally convergent series
Suppose that I = \N. If a family a_n, n \in \N, is unconditionally summable in a Hausdorff abelian topological groupX, then the series in the usual sense converges and has the same sum,
\sum_^\infty a_n = \sum_ a_n.
By nature, the definition of unconditional summability is insensitive to the order of the summation. When \sum a_n is unconditionally summable, then the series remains convergent after any permutation \sigma : \N \to \N of the set \N of indices, with the same sum,
\sum_^\infty a_ = \sum_^\infty a_n.
Conversely, if every permutation of a series \sum a_n converges, then the series is unconditionally convergent. When X is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
then unconditional convergence is also equivalent to the fact that all subseries are convergent; if X is a Banach space, this is equivalent to say that for every sequence of signs \varepsilon_n = \pm 1, the series
\sum_^\infty \varepsilon_n a_n
converges in X.
Series in topological vector spaces
If X is a topological vector space (TVS) and \left(x_i\right)_ is a (possibly uncountable) family in X then this family is summable if the limit \lim_ x_A of the net\left(x_A\right)_ exists in X, where \operatorname(I) 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 has ...
of all finite subsets of I directed by inclusion \,\subseteq\, and x_A := \sum_ x_i.
It is called absolutely summable if in addition, for every continuous seminorm p on X, the family \left(p\left(x_i\right)\right)_ is summable.
If X is a normable space and if \left(x_i\right)_ is an absolutely summable family in X, then necessarily all but a countable collection of x_i’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.
Summable families play an important role in the theory of nuclear spaces.
= Series in Banach and seminormed spaces
=
The notion of series can be easily extended to the case of a seminormed space.
If x_n is a sequence of elements of a normed space X and if x \in X then the series \sum x_n converges to x in X if the sequence of partial sums of the series \left(\sum_^N x_n\right)_^ converges to x in X; to wit,
\left\, x - \sum_^N x_n\right\, \to 0 \quad \text N \to \infty.
More generally, convergence of series can be defined in any
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
.
Specifically, in this case, \sum x_n converges to x if the sequence of partial sums converges to x.
If (X, , \cdot, ) is a seminormed space, then the notion of absolute convergence becomes:
A series \sum_ x_i of vectors in X converges absolutely if
\sum_ \left, x_i\ < +\infty
in which case all but at most countably many of the values \left, x_i\ are necessarily zero.
If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of ).
Well-ordered sums
Conditionally convergent series can be considered if I is a well-ordered set, for example, an ordinal number \alpha_0.
In this case, define by transfinite recursion:
\sum_ a_\beta = a_ + \sum_ a_\beta
and for a limit ordinal \alpha,\sum_ a_\beta = \lim_ \sum_ a_\beta
if this limit exists. If all limits exist up to \alpha_0, then the series converges.
Examples
# Given a function f : X \to Y into an abelian topological group Y, define for every a \in X,f_a(x)=
\begin
0 & x\neq a, \\
f(a) & x=a, \\
\end
a function whose Support (mathematics), support is a Singleton (mathematics), singleton \. Then
f = \sum_f_a
in the topology of pointwise convergence (that is, the sum is taken in the infinite product group Y^X).
# In the definition of partitions of unity, one constructs sums of functions over arbitrary index set I, \sum_ \varphi_i(x) = 1.
While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is ''locally finite'', that is, for every x there is a neighborhood of x in which all but a finite number of functions vanish. Any regularity property of the \varphi_i, such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
# On the first uncountable ordinal \omega_1 viewed as a topological space in the order topology, the constant function f : \left[0, \omega_1\right) \to \left[0, \omega_1\right] given by f(\alpha) = 1 satisfies \sum_f(\alpha) = \omega_1
(in other words, \omega_1 copies of 1 is \omega_1) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.
See also
* Continued fraction
* Convergence tests
* Convergent series
* Divergent series
* Infinite compositions of analytic functions
* Infinite expression (mathematics), Infinite expression
* Infinite product
* Iterated binary operation
* List of mathematical series
* Prefix sum
* Sequence transformation
* Series expansion
References
Bibliography
*Thomas John I'Anson Bromwich, Bromwich, T. J. ''An Introduction to the Theory of Infinite Series'' MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.
*
*
*
* Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).
*
*
*
*
*
*