Series (mathematics)
Encyclopedia
A series is the sum of the terms of a sequence
. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
In mathematics
, given an infinite sequence
of numbers { a_{n} }, a series is informally the result of adding all those terms together: a_{1} + a_{2} + a_{3} + · · ·. These can be written more compactly using the summation
symbol ∑. An example is the famous series from Zeno's dichotomy:
The terms of the series are often produced according to a certain rule, such as by a formula
, or by an algorithm
. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis
to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics and computer science.
of rational numbers, real numbers, complex numbers, functions
thereof, etc., the associated series is defined as the ordered formal sum.
The sequence of partial sums associated to a sequence is defined for each as the sum of the sequence from to .
By definition the series converges to a limit if and only if the associated sequence of partial sums converges to . This definition is usually written as.
More generally, if is a function
from an index set
I to a set G, then the series associated to is the formal sum of the elements over the index elements denoted by.
When the index set is the natural numbers , the function is a sequence
denoted by . A series indexed on the natural numbers is an ordered formal sum and so we rewrite as 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.
When the set is a semigroup
, the sequence of partial sums associated to a sequence is defined for each as the sum of the sequence from to .
When the semigroup is also a topological space
, then the series converges to an element if and only if the associated sequence of partial sums converges to . This definition is usually written as.
. If the limit of S_{N} is infinite or does not exist, the series is said to diverge
. When the limit of partial sums exists, it is called the sum of the series
An easy way that an infinite series can converge is if all the a_{n} are zero for n 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
: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, 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 ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. 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. Proving that the series is equal to 2 requires only elementary algebra, however. If the series is denoted S, it can be seen that
Therefore,
Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal
, as in
we are talking, in fact, just about the series
But since these series always converge to real numbers (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, it should offend no sensibilities if we make no distinction between 0.111… and ^{1}/_{9}. Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999...
for more.
on sequences. Further, this function is linear
, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference
operator, Δ. These behave as discrete analogs of integration
and differentiation
, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that
In computer science
it is known as prefix sum
.
For example, the series
is convergent, because the inequality
and a telescopic sum argument implies that the partial sums are bounded by 2.
is said to converge absolutely if the series of absolute value
s
converges. It can be proved that this is sufficient to make not only the original series converge to a limit, but also for any reordering of it to converge to the same limit.
which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series
. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the a_{n} are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.
Abel's test
is an important tool for handling semiconvergent series. If a series has the form
where the partial sums B_{N} = are bounded, λ_{n} has bounded variation, and exists:
then the series is convergent. This applies to the pointwise convergence of many trigonometric series, as in
with 0 < x < 2π. Abel's method consists in writing b_{n+1} = B_{n+1} − B_{n}, and in performing a transformation similar to integration by parts
(called summation by parts), that relates the given series to the absolutely convergent series
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
converge to ƒ(x) as N → ∞ for each x ∈ E.
A stronger notion of convergence of a series of functions is called uniform convergence. The series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,
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 termbyterm. Tests for uniform convergence include the Weierstrass' Mtest, Abel's uniform convergence test
, Dini's test.
More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere
if it converges pointwise except on a certain set of measure zero
. Other modes of convergence
depend on a different metric space
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
as N → ∞.
Many functions can be represented as Taylor series
; these are infinite series involving powers of the independent variable and are also called power series. For example, the series
converges to for all x.
In general, a power series is any series of the form
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
, and can in principle be determined from the asymptotics of the coefficients a_{n}. The convergence is uniform on closed
and bounded
(that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets
.
Historically, mathematicians such as Leonhard Euler
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.
However, the formal operation with nonconvergent series has been retained in rings of formal power series
which are studied in abstract algebra
. Formal power series are also used in combinatorics
to describe and study sequence
s that are otherwise difficult to handle; this is the method of generating function
s.
If such a series converges, then in general it does so in an annulus
rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
A Dirichlet series is one of the form
where s is a complex number
. For example, if all a_{n} are equal to 1, then the Dirichlet series is the Riemann zeta function
Like the zeta function, Dirichlet series in general play an important role in analytic number theory
. 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
outside the domain of convergence by analytic continuation
. 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 with a simple pole at 1.
This series can be directly generalized to general Dirichlet series.
s is called a trigonometric series:
The most important example of a trigonometric series is the Fourier series
of a function.
mathematician Archimedes
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
to calculate the area
under the arc of a parabola
with the summation of an infinite series, and gave a remarkably accurate approximation of π
.
The idea of an infinite series expansion of a function was conceived in India
by Madhava
of the Kerala school of astronomy and mathematics in the 14th century, who also developed precursors to the modern concepts of the power series, the Taylor series
, the Maclaurin series, rational approximations of infinite series, and infinite continued fraction
s. He discovered a number of infinite series, including the Taylor series
of the trigonometric function
s of sine
, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius
, diameter
, circumference
, angle θ, π and π/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.
In the 17th century, James Gregory
worked in the new decimal
system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series
for all functions for which they exist was provided by Brook Taylor
. Leonhard Euler
in the 18th century, developed the theory of hypergeometric series
and qseries.
in the 14th century, who developed tests of convergence
of infinite series, which his followers further developed at the Kerala school.
In Europe, however, 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
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
and Gauss
had given various criteria, and Colin Maclaurin
had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function
in such a form.
Abel
(1826) in his memoir on the binomial series
corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of and . 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
(1832), who made the first elaborate
investigation of the subject, of De Morgan
(from 1842), whose
logarithmic test DuBoisReymond (1873) and Pringsheim
(1889) have
shown to fail within a certain region; of Bertrand
(1842), Bonnet
(1843), Malmsten
(1846, 1847, the latter without integration);
Stokes
(1847), Paucker (1852), Chebyshev (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),
DuBoisReymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
limitations being pointed out by Abel, but the first to attack it
successfully were Seidel
and Stokes
(184748). 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 nonuniform
convergence, in spite of the demands of the theory of functions.
.
Semiconvergent 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 function
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.
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 (1702) and his brother Johann Bernoulli
(1701) and still
earlier by Vieta. Euler and Lagrange
simplified the subject,
as did Poinsot
, 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
(182023) 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
, Schläfli
, and
du BoisReymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were Dini, Hermite
, Halphen
,
Krause, Byerly and Appell
.
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.
, (C,k) summation, Abel summation, and Borel summation, 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
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 nonconstructive, and concerns Banach limits.
. If x_{n} is a sequence of elements of a Banach space X, then the series Σx_{n} converges to x ∈ X if the sequence of partial sums of the series tends to x; to wit,
as N → ∞.
More generally, convergence of series can be defined in any abelian
Hausdorff
topological group
. Specifically, in this case, Σx_{n} converges to x if the sequence of partial sums converges to x.
When the sum is finite, the set of i ∈ I such that a_{i} > 0 is countable. Indeed for every n ≥ 1, the set is finite, because
If I is countably infinite and enumerated as I = {i_{0}, i_{1},...} then the above defined sum satisfies
provided the value ∞ is allowed for the sum of the series.
Any sum over nonnegative reals can be understood as the integral of a nonnegative function with respect to the counting measure
, which accounts for the many similarities between the two constructions.
Hausdorff
topological group
. Let F be the collection of all finite subset
s of I. Note that F is a directed set
ordered
under inclusion with union
as join. Define the sum S of the family a as the limit
if it exists and say that the family a is unconditionally summable. Saying that the sum S is the limit of finite partial sums means that for every neighborhood V of 0 in X, there is a finite subset A_{0} of I such that
Because F is not totally ordered
, this is not a limit of a sequence
of partial sums, but rather of a net
.
For every W, neighborhood of 0 in X, there is a smaller neighborhood V such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family a_{i}, i ∈ I, form a Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A_{0} of I such that
When X is complete, a family a is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and a_{i}, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily a_{j}, j ∈ J, is also unconditionally summable in X.
When the sum of a family of nonnegative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = R.
If a family a in X is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A_{0} of I such that a_{i} ∈ W for every i not in A_{0}. If X is firstcountable
, it follows that the set of i ∈ I such that a_{i} ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).
By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑a_{n} is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,
It can be proved that the converse holds: is a series ∑a_{n} converges after any permutation, then it is unconditionally convergent. When X is complete, 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 ε_{n} = 1 or −1, the series
converges in X. If X is a Banach space, then one may define the notion of absolute convergence. A series ∑a_{n} of vectors in X converges absolutely if
If a 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 ).
α_{0}. One may define by transfinite recursion:
and for a limit ordinal α,
if this limit exists. If all limits exist up to α_{0}, then the series converges.
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, given an infinite sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of numbers { a_{n} }, a series is informally the result of adding all those terms together: a_{1} + a_{2} + a_{3} + · · ·. These can be written more compactly using the summation
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
symbol ∑. An example is the famous series from Zeno's dichotomy:
The terms of the series are often produced according to a certain rule, such as by a formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
, or by an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of welldefined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics and computer science.
Definition
For any sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of rational numbers, real numbers, complex numbers, functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
thereof, etc., the associated series is defined as the ordered formal sum.
The sequence of partial sums associated to a sequence is defined for each as the sum of the sequence from to .
By definition the series converges to a limit if and only if the associated sequence of partial sums converges to . This definition is usually written as.
More generally, if is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from an index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
I to a set G, then the series associated to is the formal sum of the elements over the index elements denoted by.
When the index set is the natural numbers , the function is a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
denoted by . A series indexed on the natural numbers is an ordered formal sum and so we rewrite as 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.
When the set is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
, the sequence of partial sums associated to a sequence is defined for each as the sum of the sequence from to .
When the semigroup is also a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, then the series converges to an element if and only if the associated sequence of partial sums converges to . This definition is usually written as.
Convergent series
A series ∑a_{n} is said to 'converge' or to 'be convergent' when the sequence S_{N} of partial sums has a finite limitLimit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
. If the limit of S_{N} is infinite or does not exist, the series is said to diverge
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
. When the limit of partial sums exists, it is called the sum of the series
An easy way that an infinite series can converge is if all the a_{n} are zero for n 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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, 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 ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. 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. Proving that the series is equal to 2 requires only elementary algebra, however. If the series is denoted S, it can be seen that
Therefore,
Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...
, as in
we are talking, in fact, just about the series
But since these series always converge to real numbers (because of what is called the completeness property
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
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, it should offend no sensibilities if we make no distinction between 0.111… and ^{1}/_{9}. Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999...
0.999...
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...
for more.
Examples
 A geometric series is one where each successive term is produced by multiplying the previous term by a constant numberMathematical constantA mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
. Example:

 In general, the geometric series
 converges if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
z < 1.
 The harmonic seriesHarmonic series (mathematics)In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
is the series

 The harmonic series is divergent.
 An alternating series is a series where terms alternate signs. Example:
 The pseries

 converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.

 converges if the sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
b_{n} converges to a limit L as n goes to infinity. The value of the series is then b_{1} − L.
Calculus and Partial Summation as an Operation on Sequences
Observe that partial summation takes as input a sequence, { a_{n} }, and gives as output another sequence, { S_{N} } – partial summation is thus a unary operationUnary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
on sequences. Further, this function is linear
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a firstdegree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
operator, Δ. These behave as discrete analogs of integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
and differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that
In computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
it is known as prefix sum
Prefix sum
In computer science, the prefix sum, or scan, of a sequence of numbers is a second sequence of numbers , the sums of prefixes of the input sequence:Parallel algorithm:A prefix sum can be calculated in parallel by the following steps....
.
Properties of series
Series are classified not only by whether they converge or diverge, but also by the properties of the terms a_{n} (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a_{n} (whether it is a real number, arithmetic progression, trigonometric function); etc.Nonnegative terms
When a_{n} is a nonnegative real number for every n, the sequence S_{N} of partial sums is nondecreasing. It follows that a series ∑a_{n} with nonnegative terms converges if and only if the sequence S_{N} of partial sums is bounded.For example, the series
is convergent, because the inequality
and a telescopic sum argument implies that the partial sums are bounded by 2.
Absolute convergence
A seriesis said to converge absolutely if the series of absolute value
Absolute value
In mathematics, the absolute value a of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of 3 is also 3...
s
converges. It can be proved that this is sufficient to make not only the original series converge to a limit, but also for any reordering of it to converge to the same limit.
Conditional convergence
A series of real or complex numbers is said to be conditionally convergent (or semiconvergent) if it is convergent but not absolutely convergent. A famous example is the alternating serieswhich is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the a_{n} are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.
Abel's test
Abel's test
In mathematics, Abel's test is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Abel...
is an important tool for handling semiconvergent series. If a series has the form
where the partial sums B_{N} = are bounded, λ_{n} has bounded variation, and exists:
then the series is convergent. This applies to the pointwise convergence of many trigonometric series, as in
with 0 < x < 2π. Abel's method consists in writing b_{n+1} = B_{n+1} − B_{n}, and in performing a transformation similar to integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
(called summation by parts), that relates the given series to the absolutely convergent series
Convergence tests
 nth term test: If lim_{n→∞ an ≠ 0 then the series diverges.}
 Comparison testComparison testIn mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
1: If ∑b_{n} is an absolutely convergentAbsolute convergenceIn mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
series such that a_{n}  ≤ C b_{n}  for some number C and for sufficiently large n , then ∑a_{n} converges absolutely as well. If ∑b_{n}  diverges, and a_{n}  ≥ b_{n}  for all sufficiently large n , then ∑a_{n} also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_{n} alternate in sign).  Comparison testComparison testIn mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
2: If ∑b_{n} is an absolutely convergent series such that a_{n+1} /a_{n}  ≤ b_{n+1} /b_{n}  for sufficiently large n , then ∑a_{n} converges absolutely as well. If ∑b_{n}  diverges, and a_{n+1} /a_{n}  ≥ b_{n+1} /b_{n}  for all sufficiently large n , then ∑a_{n} also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_{n} alternate in sign).  Ratio test: If there exists a constant C < 1 such that a_{n+1}/a_{n}<C for all sufficiently large n, then ∑a_{n} converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
 Root test: If there exists a constant C < 1 such that a_{n}^{1/n} ≤ C for all sufficiently large n, then ∑a_{n} converges absolutely.
 Integral testIntegral test for convergenceIn mathematics, the integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. An early form of the test of convergence was developed in India by Madhava in the 14th century, and by his followers at the Kerala School...
: if ƒ(x) is a positive monotone decreasing function defined on the intervalInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[ 1, ∞) with ƒ(n) = a_{n} for all n, then ∑a_{n} converges if and only if the integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
∫_{1}^{∞} ƒ(x) dx is finite.  Cauchy's condensation test: If a_{n} is nonnegative and nonincreasing, then the two series ∑a_{n} and ∑2^{k}a_{(2k)} are of the same nature: both convergent, or both divergent.
 Alternating series testAlternating series testThe alternating series test is a method used to prove that infinite series of terms converge. It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or the Leibniz criterion.A series of the form...
: A series of the form ∑(−1)^{n} a_{n} (with a_{n} ≥ 0) is called alternating. Such a series converges if the sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
a_{n} 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 seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
there is the Dini testDini testIn mathematics, the Dini and DiniLipschitz 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 :...
.
Series of functions
A series of real or complexvalued functionsconverges pointwise
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.Definition:...
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
converge to ƒ(x) as N → ∞ for each x ∈ E.
A stronger notion of convergence of a series of functions is called uniform convergence. The series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,
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
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
on a closed and bounded interval I and converge uniformly, then the series is also integrable on I and can be integrated termbyterm. Tests for uniform convergence include the Weierstrass' Mtest, Abel's uniform convergence test
Abel's uniform convergence test
In classical mathematical analysis, Abel's uniform convergence test, one of many things named after Niels Henrik Abel, is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters...
, Dini's test.
More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
if it converges pointwise except on a certain set of measure zero
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
. Other modes of convergence
Modes of convergence
In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes of convergence in the settings where they are defined...
depend on a different metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3dimensional Euclidean space...
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
as N → ∞.
Power series
Many functions can be represented as Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
; these are infinite series involving powers of the independent variable and are also called power series. For example, the series
converges to for all x.
In general, a power series is any series of the form
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
Radius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a nonnegative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
, and can in principle be determined from the asymptotics of the coefficients a_{n}. The convergence is uniform on closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
(that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets
Compact convergence
In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compactopen topology.Definition:...
.
Historically, mathematicians such as Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
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.
However, the formal operation with nonconvergent series has been retained in rings of formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
which are studied in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. Formal power series are also used in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
to describe and study sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s that are otherwise difficult to handle; this is the method of generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
s.
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 formIf such a series converges, then in general it does so in an annulus
Annulus (mathematics)
In mathematics, an annulus is a ringshaped geometric figure, or more generally, a term used to name a ringshaped object. Or, it is the area between two concentric circles...
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 is one of the form
where s is a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
. For example, if all a_{n} are equal to 1, then the Dirichlet series is the Riemann zeta function
Like the zeta function, Dirichlet series in general play an important role in analytic number theory
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 Dirichlet's introduction of Dirichlet Lfunctions to give the first proof of Dirichlet's theorem on arithmetic...
. 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
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, categories that are similar in some ways, but different in others...
outside the domain of convergence by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
. 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 with a simple pole at 1.
This series can be directly generalized to general Dirichlet series.
Trigonometric series
A series of functions in which the terms are trigonometric functionTrigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s is called a trigonometric series:
The most important example of a trigonometric series is the Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of a function.
Development of infinite series
GreekGreek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
mathematician Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
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
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 area between the nth polygon and the containing shape will...
to calculate the area
Area
Area is a quantity that expresses the extent of a twodimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
under the arc of a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
with the summation of an infinite series, and gave a remarkably accurate approximation of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
.
The idea of an infinite series expansion of a function was conceived in India
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
by Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
of the Kerala school of astronomy and mathematics in the 14th century, who also developed precursors to the modern concepts of the power series, the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
, the Maclaurin series, rational approximations of infinite series, and infinite continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s. He discovered a number of infinite series, including the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
of the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
, diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
, circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.Circumference of a circle:The circumference of a circle is the length around it....
, angle θ, π and π/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.
In the 17th century, James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions. Biography :The...
worked in the new decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
for all functions for which they exist was provided by Brook Taylor
Brook Taylor
Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series. Life and work :...
. Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in the 18th century, developed the theory of hypergeometric series
Hypergeometric series
In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...
and qseries.
Convergence criteria
The study of the convergence criteria of a series began with MadhavaMadhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
in the 14th century, who developed tests of convergence
Integral test for convergence
In mathematics, the integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. An early form of the test of convergence was developed in India by Madhava in the 14th century, and by his followers at the Kerala School...
of infinite series, which his followers further developed at the Kerala school.
In Europe, however, the investigation of the validity of infinite series is considered to begin with Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
in the 19th century. Euler had already considered the hypergeometric series
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
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions. Biography :The...
(1668). Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
and Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
had given various criteria, and Colin Maclaurin
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
in such a form.
Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.Early life:...
(1826) in his memoir on the binomial series
Binomial series
In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f = α, where is an arbitrary complex number...
corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of and . 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
Joseph Ludwig Raabe
Joseph Ludwig Raabe was a Swiss mathematician.Life:As his parents were quite poor, Raabe was forced to earn his living from a very early age by giving private lessons. He began to study mathematics in 1820 at the Polytechnicum in Vienna, Austria...
(1832), who made the first elaborate
investigation of the subject, of De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
(from 1842), whose
logarithmic test DuBoisReymond (1873) and Pringsheim
Alfred Pringsheim
Alfred Israel Pringsheim was a German mathematician and patron of the arts. He was born in Ohlau, Prussian Silesia and died in Zürich, Switzerland. Family and academic career :...
(1889) have
shown to fail within a certain region; of Bertrand
Joseph Louis François Bertrand
Joseph Louis François Bertrand was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics....
(1842), Bonnet
Pierre Ossian Bonnet
Pierre Ossian Bonnet was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the GaussBonnet theorem.Early years:...
(1843), Malmsten
Carl Johan Malmsten
Carl Johan Malmsten was a Swedish mathematician. He is notable for early research into the theory of functions of a complex variable, and for helping MittagLeffler start the journal Acta Mathematica.In 1844, he was elected a member of the Royal Swedish Academy of Sciences.References:* by...
(1846, 1847, the latter without integration);
Stokes
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
(1847), Paucker (1852), Chebyshev (1852), and Arndt
Arndt
 People with Arnd or ARent as given name :* Arent de Gelder, Dutch painter*Arnd Meier, German racecar driver*Arndt von Bohlen und Halbach*Arnd Schmitt, German fencer People with Arnd as surname :* Adolf Arndt, German politician...
(1853).
General criteria began with Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.Life:Kummer...
(1835), and have been
studied by Eisenstein (1847), Weierstrass in his various
contributions to the theory of functions, Dini (1867),
DuBoisReymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
Uniform convergence
The theory of uniform convergence was treated by Cauchy (1821), hislimitations being pointed out by Abel, but the first to attack it
successfully were Seidel
Philipp Ludwig von Seidel
Philipp Ludwig von Seidel was a German mathematician. His mother was Julie Reinhold and his father was Justus Christian Felix Seidel.Lakatos credits von Seidel with discovering, in 1847, the crucial analytic concept of uniform convergence, while analyzing an incorrect proof of Cauchy's.In 1857,...
and Stokes
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
(184748). 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 nonuniform
convergence, in spite of the demands of the theory of functions.
Semiconvergence
A series is said to be semiconvergent (or conditionally convergent) if it is convergent but not absolutely convergentAbsolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
.
Semiconvergent 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
Carl Johan Malmsten
Carl Johan Malmsten was a Swedish mathematician. He is notable for early research into the theory of functions of a complex variable, and for helping MittagLeffler start the journal Acta Mathematica.In 1844, he was elected a member of the Royal Swedish Academy of Sciences.References:* by...
(1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function
Genocchi
Angelo Genocchi
Angelo Genocchi was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi numbers are named after him.References:...
(1852) has further contributed to the theory.
Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
(1873) brought it into
prominence.
Fourier series
Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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 (1702) and his brother Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
(1701) and still
earlier by Vieta. Euler and Lagrange
Joseph Louis Lagrange
JosephLouis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
simplified the subject,
as did Poinsot
Louis Poinsot
Louis Poinsot was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple.Life:...
, Schröter, Glaisher
James Whitbread Lee Glaisher
James Whitbread Lee Glaisher son of James Glaisher, the meteorologist, was a prolific English mathematician.He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871...
, and Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.Life: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
Siméon Denis Poisson
Siméon Denis Poisson , was a French mathematician, geometer, and physicist. He however, was the final leading opponent of the wave theory of light as a member of the elite l'Académie française, but was proven wrong by AugustinJean Fresnel.Biography:...
(182023) 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
Augustin Louis Cauchy
Baron AugustinLouis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
(1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see convergence of Fourier series
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...
). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, Lipschitz
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz was a German mathematician and professor at the University of Bonn from 1864. Peter Gustav Dirichlet was his teacher. He supervised the early work of Felix Klein....
, Schläfli
Ludwig Schläfli
Ludwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...
, and
du BoisReymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were Dini, Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....
, Halphen
Georges Henri Halphen
George Henri Halphen was a French mathematician. He did his studies at École Polytechnique . He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry...
,
Krause, Byerly and Appell
Paul Émile Appell
Paul Appell , also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris...
.
Asymptotic series
Asymptotic series, otherwise asymptotic expansionAsymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...
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 summationCesàro summation
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
, (C,k) summation, Abel summation, and Borel summation, 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
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...
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 nonconstructive, and concerns Banach limits.
Series in Banach spaces
The notion of series can be easily extended to the case of a Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. If x_{n} is a sequence of elements of a Banach space X, then the series Σx_{n} converges to x ∈ X if the sequence of partial sums of the series tends to x; to wit,
as N → ∞.
More generally, convergence of series can be defined in any abelian
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
. Specifically, in this case, Σx_{n} converges to x if the sequence of partial sums converges to x.
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.Families of nonnegative numbers
When summing a family {a_{i}}, i ∈ I, of nonnegative numbers, one may defineWhen the sum is finite, the set of i ∈ I such that a_{i} > 0 is countable. Indeed for every n ≥ 1, the set is finite, because
If I is countably infinite and enumerated as I = {i_{0}, i_{1},...} then the above defined sum satisfies
provided the value ∞ is allowed for the sum of the series.
Any sum over nonnegative reals can be understood as the integral of a nonnegative function with respect to the counting measure
Counting measure
In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite....
, which accounts for the many similarities between the two constructions.
Abelian topological groups
Let a : I → X, where I is any set and X is an abelianAbelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
. Let F be the collection of all finite subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of I. Note that F is a directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
ordered
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
under inclusion with union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n. Definition :...
as join. Define the sum S of the family a as the limit
if it exists and say that the family a is unconditionally summable. Saying that the sum S is the limit of finite partial sums means that for every neighborhood V of 0 in X, there is a finite subset A_{0} of I such that
Because F is not totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
, this is not a limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
of partial sums, but rather of a net
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
.
For every W, neighborhood of 0 in X, there is a smaller neighborhood V such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family a_{i}, i ∈ I, form a Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A_{0} of I such that
When X is complete, a family a is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and a_{i}, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily a_{j}, j ∈ J, is also unconditionally summable in X.
When the sum of a family of nonnegative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = R.
If a family a in X is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A_{0} of I such that a_{i} ∈ W for every i not in A_{0}. If X is firstcountable
Firstcountable space
In topology, a branch of mathematics, a firstcountable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be firstcountable if each point has a countable neighbourhood basis...
, it follows that the set of i ∈ I such that a_{i} ≠ 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 ∈ N, is unconditionally summable in an abelian Hausdorff topological group X, then the series in the usual sense converges and has the same sum,By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑a_{n} is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,
It can be proved that the converse holds: is a series ∑a_{n} converges after any permutation, then it is unconditionally convergent. When X is complete, 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 ε_{n} = 1 or −1, the series
converges in X. If X is a Banach space, then one may define the notion of absolute convergence. A series ∑a_{n} of vectors in X converges absolutely if
If a 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 ).
Wellordered sums
Conditionally convergent series can be considered if I is a wellordered set, for example an ordinal numberOrdinal number
In set theory, an ordinal number, or just ordinal, is the order type of a wellordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
α_{0}. One may define by transfinite recursion:
and for a limit ordinal α,
if this limit exists. If all limits exist up to α_{0}, then the series converges.
Examples
Given a function f : X→Y, with Y an abelian topological group, define for every a ∈ X
a function whose supportSupport (mathematics)In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
is a singleton {a}. Then
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,
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, i.e., for every x there is a neighborhood of x in which all but a finite number of functions vanish. Any regularity property of the φ_{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 ordinalFirst uncountable ordinalIn mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals...
ω_{1} viewed as a topological space in the order topologyOrder topologyIn mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
, the constant function f: [0,ω_{1}) → [0,ω_{1}] given by f(α) = 1 satisfies
(in other words, ω_{1} copies of 1 is ω_{1}) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.
See also
 Convergent series
 Convergence testsConvergence testsIn mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series. List of tests :...
 Sequence transformation
 Infinite product
 Infinite expressionInfinite expression (mathematics)In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth...
 Continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
 Iterated binary operationIterated binary operationIn mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the...
 List of mathematical series
 Prefix sumPrefix sumIn computer science, the prefix sum, or scan, of a sequence of numbers is a second sequence of numbers , the sums of prefixes of the input sequence:Parallel algorithm:A prefix sum can be calculated in parallel by the following steps....
 Series expansionTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....