TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 generalizations ...

and its generalization,
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
. Series are used in most areas of mathematics, even for studying finite structures (such as in
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
) through
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...
s. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
,
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

and
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

. For a long time, the idea that such a potentially infinite
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: function (mathematics), fun ...

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 un ...

ical. This paradox was resolved using the concept of a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
during the 17th century.
Zeno's paradox Zeno's paradoxes are a set of philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or understanding of someo ...
of
Achilles and the tortoise Zeno's paradoxes are a set of philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or understanding of someo ...
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 or Zenon ( 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 – 2 ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
$\left(a_1,a_2,a_3,\ldots\right)$ of terms (that is, numbers,
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, 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 $a_1+a_2+a_3+\cdots,$ or, using the
summation sign In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, $\sum_^\infty a_i.$ 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 to which the terms and their finite sums belong has a notion of
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
, 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 number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

(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, $\sum_^\infty a_i = \lim_ \sum_^n a_i.$ When this limit exists, one says that the series is convergent or summable, or that the sequence $\left(a_1,a_2,a_3,\ldots\right)$ is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent. The notation $\sum_^\infty a_i$ 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 $a+b$ both the
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

—the process of adding—and its result—the ''sum'' of and . Generally, the terms of a series come from a ring, often the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
$\mathbb R$ of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or the field $\mathbb C$ of the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. In this case, the set of all series is itself a ring (and even an
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
), in which the addition consists of adding the series term by term, and the multiplication is the
Cauchy productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

# Basic properties

An infinite series or simply a series is an infinite sum, represented by an
infinite expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the form $a_0 + a_1 + a_2 + \cdots,$ where $\left(a_n\right)$ is any ordered
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of terms, such as
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s,
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, or anything else that can be (an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
). This is an expression that is obtained from the list of terms $a_0,a_1,\dots$ by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as $\sum_^ a_n .$ If an abelian group of terms has a concept of
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
(e.g., if it is a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
), then some series, the
convergent series In mathematics, a series (mathematics), series is the summation, 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 (mathematics), series that is denoted :S=a_0 +a_1+ ...
, 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 generalizations ...

, in which the group is the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or the field of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. Given a series $s=\sum_^\infty a_n$, its th partial sum is $s_k = \sum_^a_n = a_0 + a_1 + \cdots + a_k.$ By definition, the series $\sum_^ a_n$ ''converges'' to the limit (or simply ''sums'' to ), if the sequence of its partial sums has a limit . In this case, one usually writes $L = \sum_^a_n.$ 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 Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines See also

...
or to ''be convergent'' when the sequence of partial sums has a finite
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
. 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 $\sum_^\infty a_n = \lim_ s_k = \lim_ \sum_^k a_n.$ 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 $1 + \frac+ \frac+ \frac+\cdots+ \frac+\cdots.$ It is possible to "visualize" its convergence on the
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil i ... : we can imagine a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ... of length 2, with successive segment Segment or segmentation may refer to: Biology *Segmentation (biology), the division of body plans into a series of repetitive segments **Segmentation in the human nervous system *Internodal segment, the portion of a nerve fiber between two Nodes of ... s 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 some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ... . If the series is denoted , it can be seen that $S/2 = \frac = \frac+ \frac+ \frac+ \frac +\cdots.$ Therefore, $S-S/2 = 1 \Rightarrow S = 2.$ 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 A decimal representation of a non-negative real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be C ... , as in $x = 0.111\dots ,$ encodes the series $\sum_^\infty \frac.$ Since these series always converge to real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

(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 (from the Greek ἀριθμός ''arithmos'', 'number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...
; for more detail on this argument, see 0.999....

# Examples of numerical series

* A ''
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
'' 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: $1 + + + + + \cdots=\sum_^\infty = 2.$ In general, the geometric series $\sum_^\infty z^n$ converges
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
$, z, < 1$, in which case it converges to Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig ...

'' is the series $1 + + + + + \cdots = \sum_^\infty .$ The harmonic series is . * An ''
alternating series Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternatin ...
'' is a series where terms alternate signs. Examples: $1 - + - + - \cdots =\sum_^\infty =\ln(2) \quad$ (
alternating harmonic series In mathematics, the harmonic series is the divergent infinite series : \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. Its name derives from the concept of overtones, or harmonics harmonic series (music), in music: the wavelen ...

) and $-1+\frac - \frac + \frac - \frac + \cdots =\sum_^\infty \frac = -\frac$ * A
telescoping series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$\sum_^\infty (b_n-b_)$ converges if the
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''b''''n'' converges to a limit ''L''—as ''n'' goes to infinity. The value of the series is then ''b''1 − ''L''. * An '' arithmetico-geometric series'' is a generalization of the geometric series, which has
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s of the common ratio equal to the terms in an
arithmetic sequence An Arithmetic progression (AP) or arithmetic sequence is a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Eleme ...

. Example: $3 + + + + + \cdots=\sum_^\infty.$ * The ''p''-series $\sum_^\infty\frac$ converges if ''p'' > 1 and diverges for ''p'' ≤ 1, which can be shown with the integral criterion described below in
convergence tests In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. As a function of ''p'', the sum of this series is . *
Hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...
: and their generalizations (such as
basic hypergeometric series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
and
elliptic hypergeometric series In mathematics, an elliptic hypergeometric series is a series Σ''c'n'' such that the ratio ''c'n''/''c'n''−1 is an elliptic function of ''n'', analogous to generalized hypergeometric series where the ratio is a rational function of ' ...
) frequently appear in
integrable systemsIn mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantity, conserved quanti ...
and
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
. * There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series $\sum_^\infty \frac$ converges or not. The convergence depends on how well $\pi$ can be approximated with
rational numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
(which is unknown as of yet). More specifically, the values of ''n'' with large numerical contributions to the sum are the numerators of the continued fraction convergents of $\pi$, a sequence beginning with 1, 3, 22, 333, 355, 103993, ... . These are integers that are close to $n\pi$ for some integer ''n'', so that $\sin n\pi$ is close to 0 and its reciprocal is large. Alekseyev (2011) proved that if the series converges, then the irrationality measure of $\pi$ is smaller than 2.5, which is much smaller than the current known bound of 7.10320533....

## π

$\sum_^ \frac = \frac + \frac + \frac + \frac + \cdots = \frac$ $\sum_^\infty \frac = \frac - \frac + \frac - \frac + \frac - \frac + \frac - \cdots = \pi$

## Natural logarithm of 2

$\sum_^\infty \frac = \ln 2$ $\sum_^\infty \frac = \ln 2$ $\sum_^\infty \frac = 2\ln(2) -1$ $\sum_^\infty \frac = 2\ln(2) -1$ $\sum_^\infty \frac = \ln 2$ $\sum_^\infty \left(\frac+\frac\right)\frac = \ln 2$ $\sum_^\infty \frac = \ln 2$

## Natural logarithm base ''e''

$\sum_^\infty \frac = 1-\frac+\frac-\frac+\cdots = \frac$ $\sum_^\infty \frac = \frac + \frac + \frac + \frac + \frac + \cdots = e$

# Calculus and partial summation as an operation on sequences

Partial summation takes as input a sequence, (''a''''n''), and gives as output another sequence, (''S''''N''). It is thus a
unary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on sequences. Further, this function is
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

, and thus is a
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
on the
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of sequences, denoted Σ. The inverse operator is the
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
operator, denoted Δ. These behave as discrete analogues of and
differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

, 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 $\int_0^x 1\,dt = x.$ In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, it is known as
prefix sumIn computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorith ...

.

# Properties of series

Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.

## Non-negative terms

When ''an'' is a non-negative real number for every ''n'', the sequence ''SN'' of partial sums is non-decreasing. It follows that a series Σ''an'' with non-negative terms converges if and only if the sequence ''SN'' of partial sums is bounded. For example, the series $\sum_^\infty \frac$ is convergent, because the inequality $\frac1 \le \frac - \frac, \quad n \ge 2,$ and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the
Basel problem The Basel problem is a problem in mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measu ...
.

## Absolute convergence

A series $\sum_^\infty a_n$ ''converges absolutely'' if the series of
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s $\sum_^\infty \left, a_n\$ converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.

## Conditional convergence

A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series $\sum\limits_^\infty = 1 - + - + - \cdots,$ 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 Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig ...

. The
Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...
says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the $a_$ 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 In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the Convergent series, convergence of an series (mathematics), infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly ...
is an important tool for handling semi-convergent series. If a series has the form $\sum a_n = \sum \lambda_n b_n$ where the partial sums $B_ = b_ + \cdots + b_$ are bounded, $\lambda_$ has bounded variation, and $\lim \lambda_ b_$ exists: $\sup_N \left, \sum_^N b_n \ < \infty, \ \ \sum \left, \lambda_ - \lambda_n\ < \infty\ \text \ \lambda_n B_n \ \text$ then the series $\sum a_$ is convergent. This applies to the point-wise convergence of many trigonometric series, as in $\sum_^\infty \frac$ with $0 < x < 2\pi$. Abel's method consists in writing $b_=B_-B_$, and in performing a transformation similar to
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

(called
summation by parts In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), that relates the given series $\sum a_$ to the absolutely convergent series $\sum (\lambda_n - \lambda_) \, B_n.$

## Evaluation of truncation errors

The evaluation of truncation errors is an important procedure in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
(especially
validated numerics Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification (german: Zuverlässiges Rechnen) is numerics including mathematically strict error (rounding error, truncation error, discretization err ...
and
computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large Proof by exhaustion, proofs-by-exhaustion of a mathematical theorem. The ...
).

### Alternating series Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternati ...

When conditions of the alternating series test are satisfied by $S:=\sum_^\infty(-1)^m u_m$, there is an exact error evaluation. Set $s_n$ to be the partial sum $s_n:=\sum_^n(-1)^m u_m$ of the given alternating series $S$. Then the next inequality holds: $, S-s_n, \leq u_.$

### Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
is a statement that includes the evaluation of the error term when the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
is truncated.

### Hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...

By using the
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

, we can obtain the evaluation of the error term when the
hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...
is truncated.

### Matrix exponential In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

For the
matrix exponential In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
: $\exp(X) := \sum_^\infty\fracX^k,\quad X\in\mathbb^,$ the following error evaluation holds (scaling and squaring method):

# Convergence tests

There exist many tests that can be used to determine whether particular series converge or diverge. * '' n-th term test'': If $\lim_ a_n \neq 0$, then the series diverges; if $\lim_ a_n = 0$, then the test is inconclusive. * Comparison test 1 (see
Direct comparison test In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an series (mathematics ...
): If $\sum b_n$ is an
absolutely convergent In mathematics, an Series (mathematics), 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 number, real or Complex number, ...
series such that $\left\vert a_n \right\vert \leq C \left\vert b_n \right\vert$ for some number $C$ and for sufficiently large $n$, then $\sum a_n$ converges absolutely as well. If $\sum \left\vert b_n \right\vert$ diverges, and $\left\vert a_n \right\vert \geq \left\vert b_n \right\vert$ for all sufficiently large $n$, then $\sum a_n$ also fails to converge absolutely (though it could still be conditionally convergent, for example, if the $a_n$ alternate in sign). * Comparison test 2 (see
Limit comparison test In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
): If $\sum b_n$ is an absolutely convergent series such that $\left\vert \frac \right\vert \leq \left\vert \frac \right\vert$ for sufficiently large $n$, then $\sum a_n$ converges absolutely as well. If $\sum \left, b_n \$ diverges, and $\left\vert \frac \right\vert \geq \left\vert \frac \right\vert$ for all sufficiently large $n$, then $\sum a_n$ also fails to converge absolutely (though it could still be conditionally convergent, for example, if the $a_n$ alternate in sign). *
Ratio test In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero when ...
: If there exists a constant $C < 1$ such that $\left\vert \frac \right\vert < C$ for all sufficiently large $n$, then $\sum a_$ 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: If there exists a constant $C < 1$ such that $\left\vert a_ \right\vert^ \leq C$ for all sufficiently large $n$, then $\sum a_$ converges absolutely. *
Integral test In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
: if $f\left(x\right)$ is a positive
monotone decreasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...
function defined on the interval _converges_if_and_only_if_the_integral.html" ;"title=",\infty) with $f\left(n\right)=a_$ for all $n$, then $\sum a_$ converges if and only if the integral">,\infty) with $f\left(n\right)=a_$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''$a_$'' is
monotone decreasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
there is the
Dini testIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

# Series of functions

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 convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
. 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
integrableIn mathematics, integrability is a property of certain dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometric ...

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, and the Cauchy sequence, Cauchy criterion. 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 null set, measure zero. Other modes of convergence depend on a different
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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, and can in principle be determined from the asymptotics of the coefficients ''a''''n''. The convergence is uniform on closed set, closed and bounded set, bounded (that is, compact set, compact) subsets of the interior of the disc of convergence: to wit, it is Compact convergence, 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.

## 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
to describe and study
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...
s. The Hilbert–Poincaré series is a formal power series used to study graded algebras. 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 The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

, multiplication, derivative, antiderivative 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, so that the formal power series can be added term-by-term and multiplied via the
Cauchy productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. If the underlying term ring is a differential algebra, 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 (mathematics), 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.

## Dirichlet series

: A Dirichlet series 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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

. For example, if all ''a''''n'' are equal to 1, then the Dirichlet series is the Riemann zeta function $\zeta(s) = \sum_^\infty \frac.$ 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 $\C\setminus\$ with a simple pole (complex analysis), 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 functions 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of a function.

# History of the theory of infinite series

## Development of infinite series

Greek mathematics, Greek 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 Pi, π. Mathematicians from Kerala, India studied infinite series around 1350 CE. In the 17th century, James Gregory (astronomer and mathematician), 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of
hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...
and q-series.

## Convergence criteria

The investigation of the validity of infinite series is considered to begin with Carl Friedrich Gauss, 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 James Gregory (astronomer and mathematician), Gregory (1668). Leonhard Euler and Carl Friedrich Gauss, 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 (mathematics), function in such a form. Niels Henrik Abel, Abel (1826) in his memoir on the binomial series $1 + \fracx + \fracx^2 + \cdots$ 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 Joseph Ludwig Raabe, Raabe (1832), who made the first elaborate investigation of the subject, of Augustus De Morgan, De Morgan (from 1842), whose logarithmic test Paul du Bois-Reymond, DuBois-Reymond (1873) and Alfred Pringsheim, Pringsheim (1889) have shown to fail within a certain region; of Joseph Louis François Bertrand, Bertrand (1842), Pierre Ossian Bonnet, Bonnet (1843), Carl Johan Malmsten, Malmsten (1846, 1847, the latter without integration); George Gabriel Stokes, Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt (1853). General criteria began with Ernst Kummer, Kummer (1835), and have been studied by Gotthold Eisenstein, Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Ulisse Dini, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

## Uniform convergence

The theory of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Philipp Ludwig von Seidel, Seidel and George Gabriel Stokes, 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 Series (mathematics), 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 number, real or Complex number, ...
. 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 Carl Johan Malmsten, Malmsten (1847). Schlömilch (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Faulhaber's formula, Bernoulli's function $F(x) = 1^n + 2^n + \cdots + (x - 1)^n.$ Angelo Genocchi, Genocchi (1852) has further contributed to the theory. Among the early writers was Josef Hoene-Wronski, Wronski, whose "loi suprême" (1815) was hardly recognized until Arthur Cayley, Cayley (1873) brought it into prominence.

## Fourier series

Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 Franciscus Vieta, Vieta. Euler and Joseph Louis Lagrange, Lagrange simplified the subject, as did Louis Poinsot, Poinsot, Karl Schröter, Schröter, James Whitbread Lee Glaisher, Glaisher, and Ernst Kummer, 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. Siméon Denis Poisson, 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 Augustin Louis Cauchy, Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (''Journal für die reine und angewandte Mathematik, Crelle'', 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Rudolf Lipschitz, Lipschitz, Ludwig Schläfli, Schläfli, and Paul du Bois-Reymond, du Bois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Ulisse Dini, Dini, Charles Hermite, Hermite, Georges Henri Halphen, Halphen, Krause, Byerly and Paul Émile Appell, Appell.

# Generalizations

## Asymptotic series

Asymptotic series, otherwise asymptotic expansions, 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, (''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 non-constructive, and concerns Banach limits.

## 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 depends on the ordering of the index set. If $a : I \mapsto G$ is a Function (mathematics), function from an index set $I$ to a set $G,$ then the "series" associated to $a$ is the formal sum of the elements $a\left(x\right) \in G$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

denoted by $a\left(n\right) = a_n.$ 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 $\sum_a_i = \sup \left\ \in [0, +\infty].$ 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 $\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 $\left\left(a_i\right\right)_,$ from some non-empty set $I$ into is a Hausdorff space, Hausdorff Abelian group, abelian topological group $X.$ Let $\operatorname\left(I\right)$ be the collection of all Finite set, finite subsets of $I,$ with $\operatorname\left(I\right)$ viewed as a directed set, Partially ordered set, ordered under Inclusion (mathematics), inclusion $\,\subseteq\,$ with Union (set theory), union as Join (mathematics), join. The family $\left\left(a_i\right\right)_,$ is said to be if the following Limit of a net, limit, which is denoted by $\sum_ a_i$ and is called the of $\left\left(a_i\right\right)_,$ 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 $\operatorname\left(I\right)$ is not Total order, totally ordered, this is not a limit of a sequence of partial sums, but rather of a Net (mathematics), 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 $\left\left(a_i\right\right)_,$ 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 topological group, complete, a family $\left\left(a_i\right\right)_$ is unconditionally summable in $X$ if and only if the finite sums satisfy the latter Cauchy net condition. When $X$ is complete and $\left\left(a_i\right\right)_,$ is unconditionally summable in $X,$ then for every subset $J \subseteq I,$ the corresponding subfamily $\left\left(a_j\right\right)_,$ 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 $\left\left(a_i\right\right)_$ 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 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 group $X,$ 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 topological group, 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 $\varepsilon_n = \pm 1$, 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 $\left\left(x_i\right\right)_$ is a (possibly uncountable) family in $X$ then this family is summable if the limit $\lim_ x_A$ of the Net (mathematics), net $\left\left(x_A\right\right)_$ converges in $X,$ where $\operatorname\left(I\right)$ is the directed set 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 $\left\left(p\left\left(x_i\right\right)\right\right)_$ is summable. If $X$ is a normable space and if $\left\left(x_i\right\right)_$ 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 group, abelian Hausdorff space, Hausdorff topological group. Specifically, in this case, $\sum x_n$ converges to $x$ if the sequence of partial sums converges to $x.$ If $\left(X, , \cdot, \right)$ 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 $f : \left\left[0, \omega_1\right\right) \to \left\left[0, \omega_1\right\right]$ given by $f\left(\alpha\right) = 1$ 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.

* 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

*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). * * * * * *