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In mathematics, a series is often represented as the sum of a sequence of terms. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a sequence is an ordered list of objects (or events The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" That is, a series is represented as a list of numbers with addition operations between them, for example this arithmetic sequence:

1 + 2 + 3 + 4 + 5 + . Addition is the mathematical process of putting things together In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members . . + 99 + 100

In most cases of interest the terms of the sequence are produced according to a certain rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as A random number generator (often abbreviated as RNG is a computational or physical device designed to generate a sequence of Numbers or symbols that lack any

A series may be finite or infinite. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Finite series may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Analysis has its beginnings in the rigorous formulation of Calculus.

Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

\sum_{n=0}^k (an+b);

and finite geometric series, a sum of a geometric progression, which can be written as:

\sum_{n=0}^k a^{n}.

Contents

Infinite series

The sum of an infinite series a0 + a1 + a2 + … is the limit of the sequence of partial sums

S_n = a_0 + a_1 + a_2 + \cdots + a_n,

as n → ∞, if that limit exists. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found The limit of a sequence is one of the oldest concepts in Mathematical analysis. In Mathematics, a sequence is an ordered list of objects (or events If the limit exists and is finite, the series is said to converge; if it is infinite or does not exist, the series is said to diverge. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums

The easiest way that an infinite series can converge is if all the an are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

However, infinite series of nonzero terms can also converge, which resolves the mathematical side of several of Zeno's paradoxes. The simplest case of a nontrivial infinite series is perhaps

1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots.

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. In Mathematics, the real numbers may be described informally in several different ways There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

\sum_{n=0}^\infty 2^{-n}=2.

An infinite series is formally written as

\sum_{n=0}^\infty a_n

where the elements an are real (or complex) numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted We say that this series converges to S, or that its sum is S, if the limit

\lim_{N\rightarrow\infty}\sum_{n=0}^N a_n

exists and is equal to S. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" If there is no such number, then the series is said to diverge.

Formal definition

Mathematicians usually study a series as a pair of sequences: the sequence of terms of the series: a0, a1, a2, … and the sequence of partial sums S0, S1, S2, …, where Sn = a0 + a1 + … + an. The notation

\sum_{n=0}^\infty a_n

represents the above sequence of partial sums, which is always well defined, but which may or may not converge. In the case of convergence (that is, when the sequence of partial sums SN has a limit), the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may sometimes omit the limits (atop and below the sum's symbol) in the former case, although it is usually clear from the context which one is meant.

Also, different notions of convergence of such a sequence do exist (absolute convergence, summability, etc). In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the In case the elements of the sequence (and thus of the series) are not simple numbers, but, for example, functions, still more types of convergence can be considered (pointwise convergence, uniform convergence, etc. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function ; see below).

Mathematicians extend this idiom to other, equivalent notions of series. For instance, when we talk about a recurring decimal, we are talking, in fact, just about the series for which it stands (0. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is 1 + 0. 01 + 0. 001 + …). But because 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 Mathematics, the real numbers may be described informally in several different ways In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In particular, it should offend no sensibilities if we make no distinction between 0. 111… and 1/9. Less clear is the argument that 9 × 0. 111… = 0. 999… = 1, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.

History of the theory of infinite series

Development of infinite series

The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed the concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius, diameter, circumference, angle θ, π and π/4. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Theta (uppercase Θ, lowercase θ or ϑ; Θήτα is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.

In the 17th century, James Gregory also worked on infinite series and published several Maclaurin series. As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Year 1715 ( MDCCXV) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives Brook Taylor ('teɪlə(r ( 18 August 1685 &ndash 30 November 1731) was an English mathematician Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function In Mathematics, in the area of Combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a Q-analog of the common Pochhammer

Convergence criteria

The study of the convergence criteria of a series began with Madhava in the 14th century, who developed tests of convergence of infinite series, which his followers further developed at the Kerala School. In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c In Mathematics, the integral test for convergence is a method used to test infinite series of Non-negative terms for Convergence.

In Europe, however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Euler had already considered the hypergeometric series

1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots.

on which Gauss published a memoir in 1812. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Colin Maclaurin (February 1698 &ndash June 14, 1746) was a Scottish Mathematician. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Abel (1826) in his memoir on the binomial series

1 + \frac{m}{1}x + \frac{m(m-1)}{2!}x^2 + \cdots

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation In Mathematics, the binomial series generalizes the purely algebraic formula of the Binomial theorem to complex values of α 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). Joseph Ludwig Raabe (born May 15 1801 in Brody, Galicia, died January 22 1859 in Zürich, Switzerland Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. Paul David Gustav du Bois-Reymond ( December 2, 1831 &ndash April 7, 1889) was a German Mathematician who was born in Alfred Israel Pringsheim ( September 2 1850 – June 25, 1941) was a German Mathematician and artist Joseph Louis François Bertrand ( March 11, 1822 – April 5, 1900, born and died in Paris was a French Mathematician who Pierre Ossian Bonnet ( December 22, 1819 - 22 June, 1892) French Mathematician. Carl Johan Malmsten, Swedish mathematician Born April 9 1814 in Uddetorp Skara county Sweden Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist Pafnuty Lvovich Chebyshev (Пафну́тий Льво́вич Чебышёв ( –) was a Russian Mathematician. Arnd(t is a Surname, variant Arent and may refer to Adolf Arndt, German politician Alfred Arndt, German architect

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. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Ulisse Dini ( November 14, 1845 - October 28, 1918) was an Italian Mathematician and Politician born in Pringsheim's (from 1889) memoirs 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). In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence. Philipp Ludwig von Seidel ( October 24, 1821 Zweibrücken, Germany – August 13, 1896, Munich) was a German Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist 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, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the

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). Carl Johan Malmsten, Swedish mathematician Born April 9 1814 in Uddetorp Skara county Sweden Schlömilch (Zeitschrift, Vol. Oscar (Oskar Xavier Schlömilch (1823&ndash1901 was a German mathematician born in Weimar, working in Mathematical analysis. 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. In Mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum \sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \cdots + n^p Angelo Genocchi was an Italian Mathematician who specialized in Number theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence. Józef Maria Hoëne-Wroński ( August 23, 1778 – August 8, 1853) was a Polish Messianist philosopher who worked in many Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician.

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. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions 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 Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète. For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Johann Bernoulli ( Basel, 27 July 1667 - 1 January 1748 was a Swiss Mathematician. François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta, Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer. Louis Poinsot (1777 - 1859 was a French Mathematician and Physicist. James Whitbread Lee Glaisher ( 5 November 1848 - 7 December 1928) son of James Glaisher, the meteorologist was a prolific English Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician.

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. Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). In Mathematics, the question of whether the Fourier series of a Periodic function converges to the given function is researched by Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für Rudolf Otto Sigismund Lipschitz ( May 14, 1832 &ndash October 7, 1903) was a German Mathematician and professor at the Ludwig Schläfli ( 15 January, 1814 &ndash1895 was a Swiss geometer and complex analyst (at the time called Function theory DuBois-Reymond may refer to Paul David Gustav du Bois-Reymond Emil du Bois-Reymond Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell. Ulisse Dini ( November 14, 1845 - October 28, 1918) was an Italian Mathematician and Politician born in Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did George Henri Halphen ( 30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician M P Appell is the same person it stands for Monsieur Paul Appell.

Some types of infinite series

1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n}.
In general, the geometric series
\sum_{n=0}^\infty z^n
converges if and only if |z| < 1. ↔
1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots =\sum_{n=1}^\infty {1 \over n}.
The harmonic series is divergent. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series
1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}.
\sum_{n=1}^\infty\frac{1}{n^r}
converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with As a function of r, the sum of this series is Riemann's zeta function. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in
\sum_{n=1}^\infty (b_n-b_{n+1})
converges if the sequence bn converges to a limit L as n goes to infinity. In Mathematics, a telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels In Mathematics, a sequence is an ordered list of objects (or events The value of the series is then b1 − L.

Absolute convergence

Main article: absolute convergence. In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the

A series

\sum_{n=0}^\infty a_n

is said to converge absolutely if the series of absolute values

\sum_{n=0}^\infty \left|a_n\right|

converges. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.

The Riemann series theorem says that if a series is conditionally convergent then one can always find a reordering of the terms so that the reordered series diverges. In Mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem named after 19th-century German mathematician Bernhard Riemann Moreover, if the an are real and S is any real number, one can find a reordering so that the reordered series converges with limit S.

Convergence tests

Main article: convergence tests

Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + For example, the series

\sum_{n=0}^\infty\frac{x^n}{n!}

converges to ex for all x. See also radius of convergence. In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics, a sequence is an ordered list of objects (or events In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n

Dirichlet series

Main article: Dirichlet series

A Dirichlet series is one of the form

\sum_{n=1}^\infty {a_n \over n^s},

where s is a complex number. In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Generally these converge if the real part of s is greater than a number called the abscissa of convergence.

Generalizations

Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which 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. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which

Cesàro summation, (C,k) summation, Abel summation, and Borel summation provide increasingly weaker (and hence applicable to increasingly divergent series) means of defining the sums of series. In Mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums In Mathematics, a Borel summation is a generalisation of the usual notion of summation of a series

The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

Summations over arbitrary index sets

Analogous definitions may be given for sums over arbitrary index set. Let a: I → X, where I is any set and X is an abelian topological group. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Let F be the collection of all finite subsets of I. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Note that F is a directed set ordered under inclusion with union as join. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive Ordered set is used with distinct meanings in Order theory. A set with a Binary relation R on its elements that is reflexive (for In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In mathematical Order theory, join is a Binary operation on a Partially ordered set that gives the Supremum (least upper bound of its arguments We define the sum of the series as the limit

\sum_{i\in I}a_i = \lim_F\left\{\sum_{i\in A}a_i\,\bigg|A\in F\right\}

if it exists and say that the series a converges unconditionally. Thus it is the limit of all finite partial sums. Because F is not totally ordered, and because there may be uncountably many finite partial sums, this is not a limit of a sequence of partial sums, but rather of a net. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation The limit of a sequence is one of the oldest concepts in Mathematical analysis. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics

Note, however that \scriptstyle\{ i\in I : a_i > 0 \} needs to be countable for the sum to be finite. To see this, suppose it is uncountable. Then some \scriptstyle A_n = \{ i \in I : a_i > 1/n \} would be uncountable, and we can estimate the sum

\sum_{i\in I}a_i \geq \textrm{card}(A_n)/n = \infty.

This definition is insensitive to the order of the summation, so the limit will not exist for conditionally convergent series. If, however, I is a well-ordered set (for example any ordinal), one may consider the limit of partial sums of the finite initial segments

\sum_{i\in I}a_i=\lim_{n\to\infty} \sum_{i=1}^n a_i.

If this limit exists, then the series converges. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Unconditional convergence implies convergence, but not conversely, as in the case of real sequences. If X is a Banach space and I is well-ordered, then one may define the notion of absolute convergence. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis A series converges absolutely if

\lim_{n\to\infty}\sum_{i=1}^n \|a_i\|

exists. If a sequence converges absolutely then it converges unconditionally, but the converse only holds in finite dimensional Banach spaces.

Note that in some cases if the series is valued in a space that is not separable, one should consider limits of nets of partial sums over subsets of I which are not finite. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n

Real sequences

For real-valued series, an uncountable sum converges only if at most countably many terms are nonzero. Indeed, let

I_n=\left\{i\in I \,\bigg | a_i>\frac{1}{n}\right\}

be the set of indices whose terms are greater than 1/n. Each In is finite (otherwise the series would diverge). The set of indices whose terms are nonzero is the union of the In by the Archimedean principle, and the union of countably many countable sets is countable by the axiom of choice. In Abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Occasionally integrals of real functions are described as sums over the reals. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The above result shows that this interpretation should not be taken too literally. On the other hand, any sum over the reals can be understood as an integral with respect to the counting measure, which accounts for the many similarities between the two constructions. In Mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a Subset is taken to be the number

The proof goes forward in general first-countable topological vector spaces as well, such as Banach spaces; define In to be those indices whose terms are outside the n-th neighborhood of 0. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis Thus uncountable series can only be interesting if they are valued in spaces that are not first-countable.

Examples

  1. Given a function f: X→Y, with Y an abelian topological group, then define
    f_a(x)= \begin{cases} 0 & x\neq a, \\ f(a) & x=a, \\ \end{cases}
    the function whose support is a singleton {a}. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set Then
    f=\sum_{a \in X}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 This space is separable but not first countable.
  2. On the first uncountable ordinal viewed as a topological space in the order topology, the constant function f: [0,ω1] → [0,ω1] given by f(α) = 1 satisfies
    \sum_{\alpha\in[0,\omega_1]}f(\alpha) = \omega_1
    (in other words, ω1 copies of 1 is ω1) only if one takes a limit over all countable partial sums, rather than finite partial sums. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. This space is not separable.
  3. In the definition of partitions of unity, one constructs sums over arbitrary index. In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X While, formally, this requires a notion of sums of uncountable series, by construction there are only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise.

See also

References

External links

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