In

_{''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

_{''n''}), and gives as output another sequence, (''S''_{''N''}). It is thus a unary operation on sequences. Further, this function is

_{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.

_{n}'' is a non-negative real number for every ''n'', the sequence ''S_{N}'' of partial sums is non-decreasing. It follows that a series Σ''a_{n}'' with non-negative terms converges if and only if the sequence ''S_{N}'' of partial sums is bounded.
For example, the series
$$\backslash sum\_^\backslash infty\; \backslash frac$$
is convergent, because the inequality
$$\backslash frac1\; \backslash le\; \backslash frac\; -\; \backslash frac,\; \backslash quad\; n\; \backslash ge\; 2,$$
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.

By using the

_{''n''} are integrable on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the 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 measure zero. Other modes of convergence depend on a different

_{''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

sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

s that are otherwise difficult to handle, for example, using the method of

_{''n''} are equal to 1, then the Dirichlet series is the Riemann zeta function
$$\backslash zeta(s)\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$$
Like the zeta function, Dirichlet series in general play an important role in

Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

for all functions for which they exist was provided by

sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

denoted by $a(n)\; =\; a\_n.$ A series indexed on the natural numbers is an ordered formal sum and so we rewrite $\backslash sum\_$ as $\backslash sum\_^$ 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
$$\backslash sum\_^\; a\_n\; =\; a\_0\; +\; a\_1\; +\; a\_2\; +\; \backslash cdots.$$

Infinite Series Tutorial

* * {{DEFAULTSORT:Series (Mathematics) Calculus

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...

and its generalization, mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied i ...

. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

) through generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...

s. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

, computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

, statistics
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

and finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of ...

.
For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

ical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno
Zeno ( grc, Ζήνων) may refer to:
People
* Zeno (name), including a list of people and characters with the name
Philosophers
* Zeno of Elea (), philosopher, follower of Parmenides, known for his paradoxes
* Zeno of Citium (333 – 264 BC), ...

concluded that Achilles could ''never'' reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.
In modern terminology, any (ordered) infinite sequence $(a\_1,a\_2,a\_3,\backslash ldots)$ of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, o ...

like
$$a\_1+a\_2+a\_3+\backslash cdots,$$
or, using the summation sign,
$$\backslash sum\_^\backslash infty\; a\_i.$$
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, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as tends to infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...

(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,
$$\backslash sum\_^\backslash infty\; a\_i\; =\; \backslash lim\_\; \backslash sum\_^n\; a\_i.$$
When this limit exists, one says that the series is convergent or summable, or that the sequence $(a\_1,a\_2,a\_3,\backslash ldots)$ is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.
The notation $\backslash sum\_^\backslash infty\; a\_i$ 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 ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

—the process of adding—and its result—the ''sum'' of and .
Generally, the terms of a series come from a ring, often the field $\backslash mathbb\; R$ of the real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...

s or the field $\backslash mathbb\; C$ of the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. In this case, the set of all series is itself a ring (and even an associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...

), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infin ...

.
Basic properties

An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form $$a\_0\; +\; a\_1\; +\; a\_2\; +\; \backslash cdots,$$ where $(a\_n)$ is any orderedsequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

of terms, such as number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

s, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms $a\_0,a\_1,\backslash dots$ by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
$$\backslash sum\_^\; a\_n\; .$$
If an abelian group of terms has a concept of limit (e.g., if it is a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

), then some series, the convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted
:S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k.
The th partial su ...

, 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...

, in which the group is the field of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...

s or the field of complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. Given a series $s=\backslash sum\_^\backslash infty\; a\_n$, its th partial sum is
$$s\_k\; =\; \backslash sum\_^a\_n\; =\; a\_0\; +\; a\_1\; +\; \backslash cdots\; +\; a\_k.$$
By definition, the series $\backslash sum\_^\; a\_n$ ''converges'' to the limit (or simply ''sums'' to ), if the sequence of its partial sums has a limit . In this case, one usually writes
$$L\; =\; \backslash sum\_^a\_n.$$
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 or to ''be convergent'' when the sequence of partial sums has a finite limit. 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 $$\backslash sum\_^\backslash infty\; a\_n\; =\; \backslash lim\_\; s\_k\; =\; \backslash lim\_\; \backslash sum\_^k\; a\_n.$$ 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 $$1\; +\; \backslash frac+\; \backslash frac+\; \backslash frac+\backslash cdots+\; \backslash frac+\backslash cdots.$$ It is possible to "visualize" its convergence on thereal number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...

: we can imagine a line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...

of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted , it can be seen that
$$S/2\; =\; \backslash frac\; =\; \backslash frac+\; \backslash frac+\; \backslash frac+\; \backslash frac\; +\backslash cdots.$$
Therefore,
$$S-S/2\; =\; 1\; \backslash Rightarrow\; S\; =\; 2.$$
The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in
$$x\; =\; 0.111\backslash dots\; ,$$
encodes the series
$$\backslash sum\_^\backslash infty\; \backslash frac.$$
Since these series always converge to real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...

(because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the arithmetic operations
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...

; for more detail on this argument, see 0.999....
Examples of numerical series

* A ''geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...

'' is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: $$1\; +\; +\; +\; +\; +\; \backslash cdots=\backslash sum\_^\backslash infty\; =\; 2.$$In general, the geometric series

$$\backslash sum\_^\backslash infty\; z^n$$converges

if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...

$,\; z,\; <\; 1$, in which case it converges to The harmonic series is divergent.

* An ''alternating series
In mathematics, an alternating series is an infinite series of the form
\sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n
with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternati ...

'' is a series where terms alternate signs. Examples: $$1\; -\; +\; -\; +\; -\; \backslash cdots\; =\backslash sum\_^\backslash infty\; =\backslash ln(2)\; \backslash quad$$ ( alternating harmonic series) and

$$-1+\backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; =\backslash sum\_^\backslash infty\; \backslash frac\; =\; -\backslash frac$$ * Atelescoping series
In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n).
As a consequence the partial sums only consists of two terms of (a_n) after ca ...

$$\backslash sum\_^\backslash infty\; (b\_n-b\_)$$ converges if the

sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

''b''coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves va ...

s of the common ratio equal to the terms in an arithmetic sequence. Example: $$3\; +\; +\; +\; +\; +\; \backslash cdots=\backslash sum\_^\backslash infty.$$
* The ''p''-series $$\backslash sum\_^\backslash infty\backslash frac$$ converges if ''p'' > 1 and diverges for ''p'' ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of ''p'', the sum of this series is Riemann's zeta function.

* Hypergeometric series: $$\_rF\_s\; \backslash left;\; href="/html/ALL/l/\backslash begina\_1,\_a\_2,\_\backslash dotsc,\_a\_r\_\backslash \backslash \_b\_1,\_b\_2,\_\backslash dotsc,\_b\_s\_\backslash end;\_z\_\backslash right.html"\; ;"title="\backslash begina\_1,\; a\_2,\; \backslash dotsc,\; a\_r\; \backslash \backslash \; b\_1,\; b\_2,\; \backslash dotsc,\; b\_s\; \backslash end;\; z\; \backslash right">\backslash begina\_1,\; a\_2,\; \backslash dotsc,\; a\_r\; \backslash \backslash \; b\_1,\; b\_2,\; \backslash dotsc,\; b\_s\; \backslash end;\; z\; \backslash right$$and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and mathematical physics.

* There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series $$\backslash sum\_^\backslash infty\; \backslash frac$$converges or not. The convergence depends on how well $\backslash pi$ can be approximated with

rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

(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 $\backslash pi$, a sequence beginning with 1, 3, 22, 333, 355, 103993, ... . These are integers that are close to $n\backslash pi$ for some integer ''n'', so that $\backslash sin\; n\backslash pi$ is close to 0 and its reciprocal is large. Alekseyev (2011) proved that if the series converges, then the irrationality measure of $\backslash pi$ is smaller than 2.5, which is much smaller than the current known bound of 7.10320533....
Pi

$$\backslash sum\_^\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; =\; \backslash frac$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash cdots\; =\; \backslash pi$$Natural logarithm of 2

$$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash ln\; 2$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash ln\; 2$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; 2\backslash ln(2)\; -1$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; 2\backslash ln(2)\; -1$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash ln\; 2$$ $$\backslash sum\_^\backslash infty\; \backslash left(\backslash frac+\backslash frac\backslash right)\backslash frac\; =\; \backslash ln\; 2$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash ln\; 2$$Natural logarithm base ''e''

$$\backslash sum\_^\backslash infty\; \backslash frac\; =\; 1-\backslash frac+\backslash frac-\backslash frac+\backslash cdots\; =\; \backslash frac$$ $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; =\; e$$Calculus and partial summation as an operation on sequences

Partial summation takes as input a sequence, (''a''linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear re ...

, and thus is a linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...

on the vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...

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 th ...

operator, denoted Δ. These behave as discrete analogues 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 $\backslash int\_0^x\; 1\backslash ,dt\; =\; x.$
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

, it is known as prefix sum In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the sums of prefixes ( running totals) of the input sequence:
:
:
:
:...
For instance, the prefix sum ...

.
Properties of series

Series are classified not only by whether they converge or diverge, but also by the properties of the terms aNon-negative terms

When ''aGrouping

When you group a series reordering of the series does not happen, soRiemann 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 ...

does not apply. A new series will have its partial sums as subsequence of original series, which means if the original series converges, so does the new series. But for divergent series that is not true, for example 1-1+1-1+... grouped every two elements will create 0+0+0+... series, which is convergent. On the other hand, divergence of the new series means the original series can be only divergent which is sometimes useful, like in Oresme proof.
Absolute convergence

A series $$\backslash sum\_^\backslash infty\; a\_n$$ ''converges absolutely'' if the series ofabsolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

s
$$\backslash sum\_^\backslash infty\; \backslash left,\; a\_n\backslash $$
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 $$\backslash sum\backslash limits\_^\backslash infty\; =\; 1\; -\; +\; -\; +\; -\; \backslash cdots,$$ which is convergent (and its sum is equal to $\backslash ln\; 2$), but the series formed by taking the absolute value of each term is the divergent harmonic series. TheRiemann 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 is an important tool for handling semi-convergent series. If a series has the form
$$\backslash sum\; a\_n\; =\; \backslash sum\; \backslash lambda\_n\; b\_n$$
where the partial sums $B\_\; =\; b\_\; +\; \backslash cdots\; +\; b\_$ are bounded, $\backslash lambda\_$ has bounded variation, and $\backslash lim\; \backslash lambda\_\; b\_$ exists:
$$\backslash sup\_N\; \backslash left,\; \backslash sum\_^N\; b\_n\; \backslash \; <\; \backslash infty,\; \backslash \; \backslash \; \backslash sum\; \backslash left,\; \backslash lambda\_\; -\; \backslash lambda\_n\backslash \; <\; \backslash infty\backslash \; \backslash text\; \backslash \; \backslash lambda\_n\; B\_n\; \backslash \; \backslash text$$
then the series $\backslash sum\; a\_$ is convergent. This applies to the point-wise convergence of many trigonometric series, as in
$$\backslash sum\_^\backslash infty\; \backslash frac$$
with $0\; <\; x\; <\; 2\backslash pi$. Abel's method consists in writing $b\_=B\_-B\_$, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series $\backslash sum\; a\_$ to the absolutely convergent series
$$\backslash sum\; (\backslash lambda\_n\; -\; \backslash lambda\_)\; \backslash ,\; B\_n.$$
Evaluation of truncation errors

The evaluation of truncation errors is an important procedure innumerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

(especially validated numerics and computer-assisted proof
A 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 proofs-by-exhaustion of a mathematical theorem. The idea is to use ...

).

Alternating series
In mathematics, an alternating series is an infinite series of the form
\sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n
with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternati ...

When conditions of the alternating series test
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.
The test was used by Gottfried Leibniz a ...

are satisfied by $S:=\backslash sum\_^\backslash infty(-1)^m\; u\_m$, there is an exact error evaluation. Set $s\_n$ to be the partial sum $s\_n:=\backslash sum\_^n(-1)^m\; u\_m$ of the given alternating series $S$. Then the next inequality holds:
$$,\; S-s\_n,\; \backslash leq\; u\_.$$

Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

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 ...

is a statement that includes the evaluation of the error term when the Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

is truncated.
ratio
In mathematics, a ratio shows 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 the ...

, we can obtain the evaluation of the error term when the hypergeometric series is truncated.

Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

For the matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

:
$$\backslash exp(X)\; :=\; \backslash sum\_^\backslash infty\backslash fracX^k,\backslash quad\; X\backslash in\backslash mathbb^,$$
the following error evaluation holds (scaling and squaring method):
$$T\_(X)\; :=\; \backslash left;\; href="/html/ALL/l/sum\_^r\backslash frac(X/s)^j\backslash right.html"\; ;"title="sum\_^r\backslash frac(X/s)^j\backslash right">sum\_^r\backslash frac(X/s)^j\backslash right$$
Convergence tests

There exist many tests that can be used to determine whether particular series converge or diverge. * '' n-th term test'': If $\backslash lim\_\; a\_n\; \backslash neq\; 0$, then the series diverges; if $\backslash lim\_\; a\_n\; =\; 0$, then the test is inconclusive. * Comparison test 1 (see Direct comparison test): If $\backslash sum\; b\_n$ is anabsolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...

series such that $\backslash left\backslash vert\; a\_n\; \backslash right\backslash vert\; \backslash leq\; C\; \backslash left\backslash vert\; b\_n\; \backslash right\backslash vert$ for some number $C$ and for sufficiently large $n$, then $\backslash sum\; a\_n$ converges absolutely as well. If $\backslash sum\; \backslash left\backslash vert\; b\_n\; \backslash right\backslash vert$ diverges, and $\backslash left\backslash vert\; a\_n\; \backslash right\backslash vert\; \backslash geq\; \backslash left\backslash vert\; b\_n\; \backslash right\backslash vert$ for all sufficiently large $n$, then $\backslash sum\; a\_n$ 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): If $\backslash sum\; b\_n$ is an absolutely convergent series such that $\backslash left\backslash vert\; \backslash frac\; \backslash right\backslash vert\; \backslash leq\; \backslash left\backslash vert\; \backslash frac\; \backslash right\backslash vert$ for sufficiently large $n$, then $\backslash sum\; a\_n$ converges absolutely as well. If $\backslash sum\; \backslash left,\; b\_n\; \backslash $ diverges, and $\backslash left\backslash vert\; \backslash frac\; \backslash right\backslash vert\; \backslash geq\; \backslash left\backslash vert\; \backslash frac\; \backslash right\backslash vert$ for all sufficiently large $n$, then $\backslash sum\; a\_n$ 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 test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...

: If there exists a constant $C\; <\; 1$ such that $\backslash left\backslash vert\; \backslash frac\; \backslash right\backslash vert\; <\; C$ for all sufficiently large $n$, then $\backslash sum\; a\_$ 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: If there exists a constant $C\; <\; 1$ such that $\backslash left\backslash vert\; a\_\; \backslash right\backslash vert^\; \backslash leq\; C$ for all sufficiently large $n$, then $\backslash sum\; a\_$ converges absolutely.
* Integral test
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...

: if $f(x)$ is a positive monotone decreasing function defined on the interval $;\; href="/html/ALL/l/,\backslash infty)$_converges_if_and_only_if_the_integral.html" ;"title=",\infty) with $f(n)=a\_$ for all $n$, then $\backslash sum\; a\_$ converges if and only if the integral">,\infty) with $f(n)=a\_$ for all $n$, then $\backslash sum\; a\_$ converges if and only if the integral $\backslash int\_^\; f(x)\; \backslash ,\; dx$ is finite.
* Cauchy's condensation test: If $a\_$ is non-negative and non-increasing, then the two series $\backslash sum\; a\_$ and $\backslash sum\; 2^\; a\_$ are of the same nature: both convergent, or both divergent.
* Alternating series test
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.
The test was used by Gottfried Leibniz a ...

: A series of the form $\backslash sum\; (-1)^\; a\_$ (with $a\_\; >\; 0$) is called ''alternating''. Such a series converges if the sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

''$a\_$'' is monotone decreasing and converges to $0$. The converse is in general not true.
* For some specific types of series there are more specialized convergence tests, for instance for Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

there is the Dini test.
Series of functions

A series of real- or complex-valued functions $$\backslash sum\_^\backslash infty\; f\_n(x)$$ 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 $$s\_N(x)\; =\; \backslash sum\_^N\; f\_n(x)$$ converge to ''ƒ''(''x'') as ''N'' → ∞ for each ''x'' ∈ ''E''. A stronger notion of convergence of a series of functions is the uniform convergence. 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''metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

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
$$\backslash int\_E\; \backslash left,\; s\_N(x)-f(x)\backslash ^2\backslash ,dx\; \backslash to\; 0$$
as ''N'' → ∞.
Power series

: A power series is a series of the form $$\backslash sum\_^\backslash infty\; a\_n(x-c)^n.$$ TheTaylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

at a point ''c'' of a function is a power series that, in many cases, converges to the function in a neighborhood of ''c''. For example, the series
$$\backslash sum\_^\; \backslash frac$$
is the Taylor series of $e^x$ at the origin and converges to it for every ''x''.
Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''c'' in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...

, and can in principle be determined from the asymptotics of the coefficients ''a''Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
Formal power series

While many uses of power series refer to their sums, it is also possible to treat power series as ''formal sums'', meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used incombinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

to describe and study generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...

s. The Hilbert–Poincaré series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of grade ...

is a formal power series used to study graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...

s.
Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

, multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being a ...

, derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

, antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...

for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

, so that the formal power series can be added term-by-term and multiplied via the Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infin ...

. In this case the algebra of formal power series is the total algebra of the monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...

of natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...

over the underlying term ring. If the underlying term ring is a differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A na ...

, 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 $$\backslash sum\_^\backslash infty\; a\_n\; x^n.$$ 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.Dirichlet series

: ADirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyt ...

is one of the form
$$\backslash sum\_^\backslash infty\; ,$$
where ''s'' is a complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

. For example, if all ''a''analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...

. Generally a Dirichlet series converges if the real part of ''s'' is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...

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 $\backslash Complex\backslash setminus\backslash $ 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 aretrigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

s is called a trigonometric series:
$$\backslash frac12\; A\_0\; +\; \backslash sum\_^\backslash infty\; \backslash left(A\_n\backslash cos\; nx\; +\; B\_n\; \backslash sin\; nx\backslash right).$$
The most important example of a trigonometric series is the Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

of a function.
History of the theory of infinite series

Development of infinite series

Greek 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 themethod 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 ...

to calculate the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...

under the arc of a parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descrip ...

with the summation of an infinite series, and gave a remarkably accurate approximation of π.
Mathematicians from Kerala, India studied infinite series around 1350 CE.
In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Brook Taylor
Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis.
Life and work
Brook Taylor w ...

. Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

in the 18th century, developed the theory of hypergeometric series and q-series.
Convergence criteria

The investigation of the validity of infinite series is considered to begin withGauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

in the 19th century. Euler had already considered the hypergeometric series
$$1\; +\; \backslash fracx\; +\; \backslash fracx^2\; +\; \backslash cdots$$
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
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

(1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by Gregory (1668). Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

and Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orien ...

in such a form.
Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...

(1826) in his memoir on the binomial series
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x) ...

$$1\; +\; \backslash fracx\; +\; \backslash fracx^2\; +\; \backslash cdots$$
corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of $m$ and $x$. He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and
the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose
logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have
shown to fail within a certain region; of Bertrand (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),
DuBois-Reymond (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), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847–48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.Semi-convergence

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but notabsolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...

.
Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function
$$F(x)\; =\; 1^n\; +\; 2^n\; +\; \backslash cdots\; +\; (x\; -\; 1)^n.$$
Genocchi (1852) has further contributed to the theory.
Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into
prominence.
Fourier series

Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...

(1702) and his brother Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating L ...

(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
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...

'' (1822). Euler had already given the formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. Poisson (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

(1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the ...

). Dirichlet's treatment ('' Crelle'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, Lipschitz, Schläfli, and
du Bois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were Dini, Hermite, Halphen,
Krause, Byerly and Appell.
Generalizations

Asymptotic series

Asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...

, otherwise asymptotic 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 $a\; :\; I\; \backslash mapsto\; G$ is afunction
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orien ...

from an index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consist ...

$I$ to a set $G,$ then the "series" associated to $a$ is the formal sum of the elements $a(x)\; \backslash in\; G$ over the index elements $x\; \backslash in\; I$ denoted by the
$$\backslash sum\_\; a(x).$$
When the index set is the natural numbers $I=\backslash N,$ the function $a\; :\; \backslash N\; \backslash mapsto\; G$ is a Families of non-negative numbers

When summing a family $\backslash left\backslash $ of non-negative real numbers, define $$\backslash sum\_a\_i\; =\; \backslash sup\; \backslash left\backslash \; \backslash in;\; href="/html/ALL/l/,\_+\backslash infty.html"\; ;"title=",\; +\backslash infty">,\; +\backslash infty$$ When the supremum is finite then the set of $i\; \backslash in\; I$ such that $a\_i\; >\; 0$ is countable. Indeed, for every $n\; \backslash geq\; 1,$ thecardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

$\backslash left,\; A\_n\backslash $ of the set $A\_n\; =\; \backslash left\backslash $ is finite because
$$\backslash frac\; \backslash ,\; \backslash left,\; A\_n\backslash \; =\; \backslash sum\_\; \backslash frac\; \backslash leq\; \backslash sum\_\; a\_i\; \backslash leq\; \backslash sum\_\; a\_i\; <\; \backslash infty.$$
If $I$ is countably infinite and enumerated as $I\; =\; \backslash left\backslash $ then the above defined sum satisfies
$$\backslash sum\_\; a\_i\; =\; \backslash sum\_^\; a\_,$$
provided the value $\backslash infty$ 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 $a\; :\; I\; \backslash to\; X$ be a map, also denoted by $\backslash left(a\_i\backslash right)\_,$ from some non-empty set $I$ into a Hausdorff abelian topological group $X.$ Let $\backslash operatorname(I)$ be the collection of allfinite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...

subsets of $I,$ with $\backslash operatorname(I)$ viewed as a directed set, ordered under inclusion $\backslash ,\backslash subseteq\backslash ,$ with union as join.
The family $\backslash left(a\_i\backslash right)\_,$ is said to be if the following limit, which is denoted by $\backslash sum\_\; a\_i$ and is called the of $\backslash left(a\_i\backslash right)\_,$ exists in $X:$
$$\backslash sum\_\; a\_i\; :=\; \backslash lim\_\; \backslash \; \backslash sum\_\; a\_i\; =\; \backslash lim\; \backslash left\backslash $$
Saying that the sum $S\; :=\; \backslash sum\_\; a\_i$ 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
$$S\; -\; \backslash sum\_\; a\_i\; \backslash in\; V\; \backslash qquad\; \backslash text\; \backslash ;\; A\; \backslash supseteq\; A\_0.$$
Because $\backslash operatorname(I)$ is not totally ordered, this is not a limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...

of partial sums, but rather of a net.
For every neighborhood $W$ of the origin in $X,$ there is a smaller neighborhood $V$ such that $V\; -\; V\; \backslash subseteq\; W.$ It follows that the finite partial sums of an unconditionally summable family $\backslash left(a\_i\backslash right)\_,$ form a , that is, for every neighborhood $W$ of the origin in $X,$ there exists a finite subset $A\_0$ of $I$ such that
$$\backslash sum\_\; a\_i\; -\; \backslash sum\_\; a\_i\; \backslash in\; W\; \backslash qquad\; \backslash text\; \backslash ;\; A\_1,\; A\_2\; \backslash supseteq\; A\_0,$$
which implies that $a\_i\; \backslash in\; W$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_0$ (by taking $A\_1\; :=\; A\_0\; \backslash cup\; \backslash $ and $A\_2\; :=\; A\_0$).
When $X$ is complete, a family $\backslash left(a\_i\backslash right)\_$ is unconditionally summable in $X$ if and only if the finite sums satisfy the latter Cauchy net condition. When $X$ is complete and $\backslash left(a\_i\backslash right)\_,$ is unconditionally summable in $X,$ then for every subset $J\; \backslash subseteq\; I,$ the corresponding subfamily $\backslash left(a\_j\backslash right)\_,$ 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 $X\; =\; \backslash R.$
If a family $\backslash left(a\_i\backslash right)\_$ in $X$ is unconditionally summable then for every neighborhood $W$ of the origin in $X,$ there is a finite subset $A\_0\; \backslash subseteq\; I$ such that $a\_i\; \backslash in\; W$ for every index $i$ not in $A\_0.$ If $X$ is a first-countable space then it follows that the set of $i\; \backslash in\; I$ such that $a\_i\; \backslash neq\; 0$ is countable. This need not be true in a general abelian topological group (see examples below).
Unconditionally convergent series

Suppose that $I\; =\; \backslash N.$ If a family $a\_n,\; n\; \backslash in\; \backslash N,$ is unconditionally summable in a Hausdorff abelian topological group $X,$ then the series in the usual sense converges and has the same sum, $$\backslash sum\_^\backslash infty\; a\_n\; =\; \backslash sum\_\; a\_n.$$ By nature, the definition of unconditional summability is insensitive to the order of the summation. When $\backslash sum\; a\_n$ is unconditionally summable, then the series remains convergent after anypermutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

$\backslash sigma\; :\; \backslash N\; \backslash to\; \backslash N$ of the set $\backslash N$ of indices, with the same sum,
$$\backslash sum\_^\backslash infty\; a\_\; =\; \backslash sum\_^\backslash infty\; a\_n.$$
Conversely, if every permutation of a series $\backslash sum\; a\_n$ converges, then the series 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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...

, this is equivalent to say that for every sequence of signs $\backslash varepsilon\_n\; =\; \backslash pm\; 1$, the series
$$\backslash sum\_^\backslash infty\; \backslash varepsilon\_n\; a\_n$$
converges in $X.$
Series in topological vector spaces

If $X$ is a topological vector space (TVS) and $\backslash left(x\_i\backslash right)\_$ is a (possiblyuncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...

) family in $X$ then this family is summable if the limit $\backslash lim\_\; x\_A$ of the net $\backslash left(x\_A\backslash right)\_$ exists in $X,$ where $\backslash operatorname(I)$ is the directed set of all finite subsets of $I$ directed by inclusion $\backslash ,\backslash subseteq\backslash ,$ and $x\_A\; :=\; \backslash sum\_\; x\_i.$
It is called absolutely summable if in addition, for every continuous seminorm $p$ on $X,$ the family $\backslash left(p\backslash left(x\_i\backslash right)\backslash right)\_$ is summable.
If $X$ is a normable space and if $\backslash left(x\_i\backslash right)\_$ 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 $x\; \backslash in\; X$ then the series $\backslash sum\; x\_n$ converges to $x$ in $X$ if the sequence of partial sums of the series $\backslash left(\backslash sum\_^N\; x\_n\backslash right)\_^$ converges to $x$ in $X$; to wit, $$\backslash left\backslash ,\; x\; -\; \backslash sum\_^N\; x\_n\backslash right\backslash ,\; \backslash to\; 0\; \backslash quad\; \backslash text\; N\; \backslash to\; \backslash infty.$$ More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, $\backslash sum\; x\_n$ converges to $x$ if the sequence of partial sums converges to $x.$ If $(X,\; ,\; \backslash cdot,\; )$ is a seminormed space, then the notion of absolute convergence becomes: A series $\backslash sum\_\; x\_i$ of vectors in $X$ converges absolutely if $$\backslash sum\_\; \backslash left,\; x\_i\backslash \; <\; +\backslash infty$$ in which case all but at most countably many of the values $\backslash left,\; x\_i\backslash $ 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 awell-ordered
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...

set, for example, an ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least ...

$\backslash alpha\_0.$
In this case, define by transfinite recursion
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for ...

:
$$\backslash sum\_\; a\_\backslash beta\; =\; a\_\; +\; \backslash sum\_\; a\_\backslash beta$$
and for a limit ordinal $\backslash alpha,$
$$\backslash sum\_\; a\_\backslash beta\; =\; \backslash lim\_\; \backslash sum\_\; a\_\backslash beta$$
if this limit exists. If all limits exist up to $\backslash alpha\_0,$ then the series converges.
Examples

# Given a function $f\; :\; X\; \backslash to\; Y$ into an abelian topological group $Y,$ define for every $a\; \backslash in\; X,$ $$f\_a(x)=\; \backslash begin\; 0\; \&\; x\backslash neq\; a,\; \backslash \backslash \; f(a)\; \&\; x=a,\; \backslash \backslash \; \backslash end$$a function whose support is a singleton $\backslash .$ Then

$$f\; =\; \backslash sum\_f\_a$$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 $I,$ $$\backslash sum\_\; \backslash varphi\_i(x)\; =\; 1.$$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 $\backslash varphi\_i,$ 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 $\backslash omega\_1$ viewed as a topological space in theorder topology
In 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.
If ''X'' is a totally ordered set, t ...

, the constant function $f\; :\; \backslash left;\; href="/html/ALL/l/,\_\backslash omega\_1\backslash right)\_\backslash to\_\backslash left[0,\_\backslash omega\_1\backslash right.html"\; ;"title=",\; \backslash omega\_1\backslash right)\; \backslash to\; \backslash left[0,\; \backslash omega\_1\backslash right">,\; \backslash omega\_1\backslash right)\; \backslash to\; \backslash left[0,\; \backslash omega\_1\backslash right$(in other words, $\backslash omega\_1$ copies of 1 is $\backslash omega\_1$) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.

See also

* Continued fraction * Convergence tests * Convergent series * Divergent series * Infinite compositions of analytic functions * Infinite expression (mathematics), Infinite expression * Infinite product * Iterated binary operation * List of mathematical series *Prefix sum In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the sums of prefixes ( running totals) of the input sequence:
:
:
:
:...
For instance, the prefix sum ...

* Sequence transformation
* Series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division) ...

References

Bibliography

* 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). * * * * * *External links

*Infinite Series Tutorial

* * {{DEFAULTSORT:Series (Mathematics) Calculus