Generalized series (mathematics)
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In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of
calculus 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 arithm ...
and its generalization,
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. 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 r ...
,
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 (includi ...
, statistics and finance. 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 Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in pluralit ...
illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.
Zeno Zeno ( grc, Ζήνων) may refer to: People * Zeno (name), including a list of people and characters with the name Philosophers * Zeno of Elea (), philosopher, follower of Parmenides, known for his paradoxes * Zeno of Citium (333 – 264 BC), ...
concluded that Achilles could ''never'' reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. In modern terminology, any (ordered)
infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
(a_1,a_2,a_3,\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+\cdots, or, using the
summation sign In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
, \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, 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 (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 (a_1,a_2,a_3,\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 \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—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 grass ...
\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. Every ...
s or the field \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 fo ...
s. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the
Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...
.


Basic properties

An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form a_0 + a_1 + a_2 + \cdots, where (a_n) is any ordered
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
). This is an expression that is obtained from the list of terms 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 (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 set ...
), then some series, the
convergent series In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial ...
, can be interpreted as having a value in , called the ''sum of the series''. This includes the common cases from
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, in which the group is the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. Given a series 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 *CONVERGE CFD s ...
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 \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 In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
. If the series is denoted , it can be seen that 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 of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if a ...
, as in x = 0.111\dots , encodes the series \sum_^\infty \frac. Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the
arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
; 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 + + + + + \cdots=\sum_^\infty = 2.

In general, the geometric series

\sum_^\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 b ...
, z, < 1, in which case it converges to harmonic series'' is the series 1 + + + + + \cdots = \sum_^\infty .

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 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 infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, wher ...
) and

-1+\frac - \frac + \frac - \frac + \cdots =\sum_^\infty \frac = -\frac * A
telescoping 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 c ...
\sum_^\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 calle ...
''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 coefficients of the common ratio equal to the terms in an
arithmetic sequence An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
. 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, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n. List of tests Limit of the summand If t ...
. As a function of ''p'', the sum of this series is Riemann's zeta function.

* Hypergeometric series: _rF_s \left \begina_1, a_2, \dotsc, a_r \\ b_1, b_2, \dotsc, b_s \end; z \right:= \sum_^ \frac z^n

and their generalizations (such as

basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...
and elliptic hypergeometric series) frequently appear in integrable systems 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, 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 rat ...
(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....


Pi

\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, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
on sequences. Further, this function is
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 ...
, and thus is a linear operator on the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 t ...
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 \int_0^x 1\,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 (includi ...
, 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 sums ...
.


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 with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
.


Grouping

When you group a series reordering of the series does not happen, so
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 ...
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 \sum_^\infty a_n ''converges absolutely'' if the series of absolute values \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. 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 convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test &nd ...
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, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
(called
summation by parts In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformat ...
), 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 manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
(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 er ...
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 a ...
).


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 alternatin ...

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:=\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, 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 th ...
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.


Hypergeometric series

By using the
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 give ...

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 give ...
: \exp(X) := \sum_^\infty\fracX^k,\quad X\in\mathbb^, the following error evaluation holds (scaling and squaring method): T_(X) := \left sum_^r\frac(X/s)^j\rights,\quad \, \exp(X)-T_(X)\, \leq\frac\exp(\, X\, ).


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 infinite series ...
): If \sum b_n is an
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
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, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement Suppose that we have two series \Sigma_n a_n and \Sigma_n b_n ...
): 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 test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
: 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, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity :\limsup_\sqrt where a_n are the terms of the series, and states that the series converges absolutely if ...
: 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, 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 \sum_a__converges_if_and_only_if_the_integral.html" ;"title=",\infty) with f(n)=a_ for all n, then \sum a_ converges if and only if the integral">,\infty) with f(n)=a_ for all n, then \sum a_ converges if and only if the integral \int_^ f(x) \, dx is finite. * Cauchy's condensation test: If a_ is non-negative and non-increasing, then the two series \sum a_ and \sum 2^ a_ are of the same nature: both convergent, or both divergent. *
Alternating series test 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 \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 calle ...
''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 there is the
Dini test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition L ...
.


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 convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
. A series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum, , s_N(x) - f(x), can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''. Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ''ƒ''''n'' are
integrable In 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 quantities, or first ...
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 In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test &nda ...
, 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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
if it converges pointwise except on a certain set of
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
. Other
modes of convergence In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, se ...
depend on a different
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set ''E'' to a limit function ''ƒ'' provided \int_E \left, s_N(x)-f(x)\^2\,dx \to 0 as ''N'' → ∞.


Power series

: A power series is a series of the form \sum_^\infty a_n(x-c)^n. The
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
at a point ''c'' of a function is a power series that, in many cases, converges to the function in a neighborhood of ''c''. For example, the series \sum_^ \frac is the Taylor series of e^x at the origin and converges to it for every ''x''. Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''c'' in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
, and can in principle be determined from the asymptotics of the coefficients ''a''''n''. The convergence is uniform on closed and bounded (that is,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets. Historically, mathematicians such as
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 ma ...
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 to describe and study
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s that are otherwise difficult to handle, for example, using the method of generating functions. 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 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, multiplication,
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 symbolicall ...
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 product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...
. In this case the algebra of formal power series is the total algebra of the
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
of natural numbers over the underlying term ring. If the underlying term ring is a
differential algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.


Laurent series

Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form \sum_^\infty a_n x^n. If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.


Dirichlet series

: A
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analy ...
is one of the form \sum_^\infty , where ''s'' is a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
. For example, if all ''a''''n'' are equal to 1, then the Dirichlet series is the Riemann zeta function \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 In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
outside the domain of convergence by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
. For example, the Dirichlet series for the zeta function converges absolutely when Re(''s'') > 1, but the zeta function can be extended to a holomorphic function defined on \Complex\setminus\ with a simple
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
at 1. This series can be directly generalized to general Dirichlet series.


Trigonometric series

A series of functions in which the terms are
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s is called a trigonometric series: \frac12 A_0 + \sum_^\infty \left(A_n\cos nx + B_n \sin nx\right). The most important example of a trigonometric series is the Fourier series of a function.


History of the theory of infinite series


Development of infinite series

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
to calculate the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
under the arc of a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
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 Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
. In 1715, a general method for constructing the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
for all functions for which they exist was provided by Brook Taylor.
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 ma ...
in the 18th century, developed the theory of hypergeometric series and
q-series In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer sym ...
.


Convergence criteria

The investigation of the validity of infinite series is considered to begin with
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 refer ...
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 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 w ...
(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 in ma ...
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 refer ...
had given various criteria, and
Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
by his expansion of a complex
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 + \fracx + \fracx^2 + \cdots corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. He showed the necessity of considering the subject of continuity in questions of convergence. Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of
De Morgan De Morgan or de Morgan is a surname, and may refer to: * Augustus De Morgan (1806–1871), British mathematician and logician. ** De Morgan's laws (or De Morgan's theorem), a set of rules from propositional logic. ** The De Morgan Medal, a trien ...
(from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842),
Bonnet A Bonnet is a variety of headgear, hat or cap Specific types of headgear referred to as "bonnets" may include Scottish * Blue bonnet, a distinctive woollen cap worn by men in Scotland from the 15th-18th centuries And its derivations: ** Fea ...
(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 Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.


Uniform convergence

The theory of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847–48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.


Semi-convergence

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
. Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function F(x) = 1^n + 2^n + \cdots + (x - 1)^n. Genocchi (1852) has further contributed to the theory. Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.


Fourier series

Fourier series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
(1702) and his brother
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
(1701) and still earlier by Vieta. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer. Fourier (1807) set for himself a different problem, to expand a given function of ''x'' in terms of the sines or cosines of multiples of ''x'', a problem which he embodied in his '' Théorie analytique de la chaleur'' (1822). Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (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 w ...
(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 th ...
). 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 expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
s, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.


Divergent series

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A
summability method In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...
is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include
Cesàro summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean ) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
, (''C'',''k'') summation,
Abel summation In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...
, and
Borel summation In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several var ...
, 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 limit In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\in ...
s.


Summations over arbitrary index sets

Definitions may be given for sums over an arbitrary index set I. There are two main differences with the usual notion of series: first, there is no specific order given on the set I; second, this set I may be uncountable. The notion of convergence needs to be strengthened, because the concept of
conditional convergence In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Definition More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if \lim_\,\s ...
depends on the ordering of the index set. If a : I \mapsto G is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from an
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
I to a set G, then the "series" associated to a is the
formal sum In mathematics, a formal sum, formal series, or formal linear combination may be: *In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. *In linear algebra, an ...
of the elements a(x) \in G over the index elements x \in I denoted by the \sum_ a(x). When the index set is the natural numbers I=\N, the function a : \N \mapsto G is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
denoted by a(n) = a_n. A series indexed on the natural numbers is an ordered formal sum and so we rewrite \sum_ as \sum_^ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers \sum_^ a_n = a_0 + a_1 + a_2 + \cdots.


Families of non-negative numbers

When summing a family \left\ of non-negative real numbers, define \sum_a_i = \sup \left\ \in , +\infty When the supremum is finite then the set of i \in I such that a_i > 0 is countable. Indeed, for every n \geq 1, the cardinality \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 In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
, which accounts for the many similarities between the two constructions.


Abelian topological groups

Let a : I \to X be a map, also denoted by \left(a_i\right)_, from some non-empty set I into a Hausdorff abelian
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
X. Let \operatorname(I) be the collection of all
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
subsets of I, with \operatorname(I) viewed as a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
, ordered under
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
\,\subseteq\, with
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
as
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
. The family \left(a_i\right)_, is said to be if the following limit, which is denoted by \sum_ a_i and is called the of \left(a_i\right)_, exists in X: \sum_ a_i := \lim_ \ \sum_ a_i = \lim \left\ Saying that the sum S := \sum_ a_i is the limit of finite partial sums means that for every neighborhood V of the origin in X, there exists a finite subset A_0 of I such that S - \sum_ a_i \in V \qquad \text \; A \supseteq A_0. Because \operatorname(I) is not
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
, this is not a
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limi ...
of partial sums, but rather of a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
. For every neighborhood W of the origin in X, there is a smaller neighborhood V such that V - V \subseteq W. It follows that the finite partial sums of an unconditionally summable family \left(a_i\right)_, form a , that is, for every neighborhood W of the origin in X, there exists a finite subset A_0 of I such that \sum_ a_i - \sum_ a_i \in W \qquad \text \; A_1, A_2 \supseteq A_0, which implies that a_i \in W for every i \in I \setminus A_0 (by taking A_1 := A_0 \cup \ and A_2 := A_0). When X is complete, a family \left(a_i\right)_ is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and \left(a_i\right)_, is unconditionally summable in X, then for every subset J \subseteq I, the corresponding subfamily \left(a_j\right)_, is also unconditionally summable in X. When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = \R. If a family \left(a_i\right)_ in X is unconditionally summable then for every neighborhood W of the origin in X, there is a finite subset A_0 \subseteq I such that a_i \in W for every index i not in A_0. If X is a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
then it follows that the set of i \in I such that a_i \neq 0 is countable. This need not be true in a general abelian topological group (see examples below).


Unconditionally convergent series

Suppose that I = \N. If a family a_n, n \in \N, is unconditionally summable in a Hausdorff abelian topological 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 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) and \left(x_i\right)_ is a (possibly
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
) family in X then this family is summable if the limit \lim_ x_A of the
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
\left(x_A\right)_ exists in X, where \operatorname(I) is the
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
of all finite subsets of I directed by inclusion \,\subseteq\, and x_A := \sum_ x_i. It is called absolutely summable if in addition, for every continuous seminorm p on X, the family \left(p\left(x_i\right)\right)_ is summable. If X is a normable space and if \left(x_i\right)_ is an absolutely summable family in X, then necessarily all but a countable collection of x_i’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms. Summable families play an important role in the theory of
nuclear space In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
s.


= 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 Hausdorff
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
. Specifically, in this case, \sum x_n converges to x if the sequence of partial sums converges to x. If (X, , \cdot, ) is a seminormed space, then the notion of absolute convergence becomes: A series \sum_ x_i of vectors in X converges absolutely if \sum_ \left, x_i\ < +\infty in which case all but at most countably many of the values \left, x_i\ are necessarily zero. If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of ).


Well-ordered sums

Conditionally convergent series can be considered if I is a
well-ordered 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-or ...
set, for example, an ordinal number \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 a ...
: \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 is a

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
\. Then

f = \sum_f_a

in the

topology of pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
(that is, the sum is taken in the infinite product group Y^X).

# In the definition of
partitions of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
, 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 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, th ...
, the constant function f : \left , \omega_1\right) \to \left[0, \omega_1\right/math> given by f(\alpha) = 1 satisfies \sum_f(\alpha) = \omega_1

(in other words, \omega_1 copies of 1 is \omega_1) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.


See also

* Continued fraction * Convergence tests * Convergent series * Divergent series * Infinite compositions of analytic functions * Infinite expression (mathematics), Infinite expression * Infinite product *
Iterated binary operation In mathematics, an iterated binary operation is an extension of a binary operation on a set ''S'' to a function on finite sequences of elements of ''S'' through repeated application. Common examples include the extension of the addition operation ...
*
List of mathematical series This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. *Here, 0^0 is taken to have the value 1 *\ denotes the fractional part of x *B_n(x) is a Bernoul ...
*
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 sums ...
* 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 divisi ...


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