NationMaster - Encyclopedia: Series (mathematics)

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 addition operations between them, for example this arithmetic sequence: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For evaluation of sums in closed form see evaluating sums. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... This is a page about mathematics. ...

1 + 2 + 3 + 4 + 5 + ... + 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, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ... A random number generator is a computational or physical device designed to generate a sequence of elements (usually numbers), such that the sequence can be used as a random one. ...

A series may be finite or infinite. 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. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... 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: // Definition In mathematics, an arithmetic series is the sum of the components of an arithmetic progression. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

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

and finite geometric series, a sum of a geometric progression, which can be written as: Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ...

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

1 Infinite series
- 1.1 Formal definition
2 History of the theory of infinite series
3 Some types of infinite series
4 Absolute convergence
5 Convergence tests
6 Power series
7 Dirichlet series
8 Generalizations
- 8.1 Summations over arbitrary index sets
  - 8.1.1 Real sequences
  - 8.1.2 Examples
9 See also
10 References

Infinite series

The sum of an infinite series a₀ + a₁ + a₂ + … is the limit of the sequence of partial sums The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

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

as n → ∞, if the partial sums converge to a finite value. If so, the series is said to converge; otherwise the series is said to diverge. In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a divergent series is an infinite series that does not converge. ...

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

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 â€œArrow paradoxâ€ redirects here. ...

$1+frac{1}{2}+frac{1}{4}+frac{1}{8}+frac{1}{16}+cdots$

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. 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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 a_n are real (or complex) numbers. We say that this series converges to S, or that its sum is S, if the limit In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

$lim_{Nrightarrowinfty}sum_{n=0}^N a_n$

exists and is equal to S. 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 S_n = a₀ + a₁ + … + a_n. The notation

$sum_{n=0}^infty a_n$

represents then a priori this pair of sequences, which is always well defined, but which may or may not converge. In the case of convergence, i.e., if the sequence of partial sums S_N 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. The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ...

Also, different notions of convergence of such a sequence do exist (absolute convergence, summability, etc). 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.; see below). In mathematics, a series is a sum of a sequence of terms. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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 (0.1 + 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 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. A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... Please refer to Real vs. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the recurring decimal 0. ...

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. 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. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Madhavan (à´®à´¾à´§à´µà´¨àµ) of Sangamagramam (1350â€“1425) was a prominent mathematician-astronomer from Kerala, India. ... This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... As the degree of the Taylor series rises, it approaches the correct function. ... As the degree of the taylor series rises, it approaches the correct function. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... As the degree of the Taylor series rises, it approaches the correct function. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... DIAMETER is an AAA protocol (Authentication, Authorization and Accounting) succeeding its predecessor RADIUS. // The name is a pun on the RADIUS protocol, which is the predecessor (a diameter is twice the radius). ... The circumference is the distance around a closed curve. ... Note: A theta probe is a device for measuring soil moisture. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†ÎÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ...

In the 17th century, James Gregory also worked on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series. (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... James Gregory For other people with the same name, see James Gregory. ... As the degree of the taylor series rises, it approaches the correct function. ... Year 1715 (MDCCXV) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 11-day slower Julian calendar). ... As the degree of the Taylor series rises, it approaches the correct function. ... Brook Taylor (August 18, 1685 â€“ November 30, 1731) was an English mathematician. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ...

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 toward a fixed or equilibrium state. ... 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. Euler had already considered the hypergeometric series Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

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

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... James Gregory For other people with the same name, see James Gregory. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Colin Maclaurin Colin Maclaurin (February, 1698 - June 14, 1746) was a Scottish mathematician. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

Abel (1826) in his memoir on the binomial series Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ... In mathematics, the binomial series generalizes the purely algebraic binomial theorem. ...

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

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of $m$ and $x$ . 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), Tchebichef (1852), and Arndt (1853). Joseph Ludwig Raabe (born May 15, 1801 in Brody, Galicia, died January 22, 1859 in ZÃ¼rich, Switzerland) was a Swiss mathematician. ... Augustus De Morgan (June 27, 1806 â€“ March 18, 1871) was an Indian-born British mathematician and logician. ... Paul David Gustav du Bois-Reymond (December 2, 1831 - April 7, 1889) was a mathematician who was born in Berlin, Germany and died in Freiburg, Germany. ... Pringsheim may mean: Alfred Pringsheim (1850-1941), mathematician Nathanael Pringsheim (1823-1894), German botanist This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Bertrand is the name of several persons: Alexander Bertrand (1820-1902), French archeologist. ... A bonnet the name of different types of headwear for men and women. ... Stokes may refer to the following: George Gabriel Stokes (1819â€“1903), Irish mathematician and physicist Stokes theorem Navier-Stokes equations Stokes law Stokes relations Stokes (unit), the cgs unit of kinematic viscosity Stokes radius Stokes shift Stokes (lunar crater) Stokes (crater on Mars) Alexander Rawson Stokes (Alec), 1919â€“2003, English... Pafnuty Lvovich Chebyshev Pafnuty Lvovich Chebyshev (Пафнутий Львович Чебышёв) (May 4, 1821 - November 26, 1894) was a Russian mathematician. ... Arnd(t) is a surname, variant Arent and may refer to: Alfred Arndt, German architect Bettina Arndt, Australian sex therapist Chip Arndt, American reality show contestant Denis Arndt, actor Ernst Moritz Arndt, German author/poet Felix Arndt, American pianist Ingrid Arndt-Brauer, German politician Johann Arndt, German theologian Judith Arndt...

General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory. Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Dini may be Dino Dini, games developer Lamberto Dini, politician Ulisse Dini, mathematician This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Paul David Gustav du Bois-Reymond (December 2, 1831 - April 7, 1889) was a mathematician who was born in Berlin, Germany and died in Freiburg, Germany. ...

Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions. In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... Philipp Ludwig von Seidel (1821â€“1896) was a German mathematician. ... Sir George Gabriel Stokes, 1st Baronet (13 August 1819â€“1 February 1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics (including Stokes theorem). ...

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 is a sum of a sequence of terms. ...

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 Carl Johan Malmsten, Swedish mathematican. ... Oscar (Oskar) Xavier SchlÃ¶milch (1823â€“1901) was a German mathematician, born in Weimar, working in mathematical analysis. ... In mathematics, Faulhabers formula, named after Johann Faulhaber, expresses the sum as a (p + 1)th-degree polynomial function of x, the coefficients involving Bernoulli numbers. ...

$F(x) = 1^n + 2^n + cdots + (x - 1)^n.,$

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. Josef Hoëné-Wronski (August 23, 1778 - August 8, 1853), born Josef Hoëné, was a Polish eccentric philosopher of mathematics. ... 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. 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. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... FranÃ§ois ViÃ¨te. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Louis Poinsot (1777 - 1859) was a French mathematician and physicist. ... James Whitbread Lee Glaisher (5 November 1848 - 7 December 1928) was a prolific British mathematician. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) 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. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell. SimÃ©on Poisson. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis, a branch of pure mathematics. ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ... Rudolf Otto Sigismund Lipschitz (May 14, 1832 â€“ October 7, 1903) was a German mathematician and Professor at the University of Bonn from 1864. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Paul David Gustav du Bois-Reymond (December 2, 1831 - April 7, 1889) was a mathematician who was born in Berlin, Germany and died in Freiburg, Germany. ... Ulisse Dini (Born November 14, 1845 in Pisa, Italy-Died October 28, 1918 in Pisa, Italy) was a mathematician and politician. ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... George Henri Halphen (30 October 1844 - 23 May 1889) was a French mathematician. ... Paul Ã‰mile Appell (September 27, 1855, Strasbourg - October 23, 1930) was the French mathematician. ...

Some types of infinite series

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:

$1 + {1 over 2} + {1 over 4} + {1 over 8} + {1 over 16} + cdots=sum_{n=0}^infty{1 over 2^n}.$

In general, the geometric series

$sum_{n=0}^infty z^n$

converges if and only if |z| < 1.

The harmonic series is the series

$1 + {1 over 2} + {1 over 3} + {1 over 4} + {1 over 5} + cdots =sum_{n=1}^infty {1 over n}.$

An alternating series is a series where terms alternate signs. Example:

$1 - {1 over 2} + {1 over 3} - {1 over 4} + {1 over 5} - cdots =sum_{n=1}^infty (-1)^{n+1} {1 over n}.$

The series

$sum_{n=1}^inftyfrac{1}{n^r}$

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

A telescoping series

$sum_{n=1}^infty (b_n-b_{n+1})$

converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ − L.

In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... See harmonic series (music) for the (related) musical concept. ... In mathematics, an alternating series is an infinite series of the form with an â‰¥ 0. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with a succeeding or preceding term. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

Absolute convergence

Main article: absolute convergence.

A series In mathematics, a series is a sum of a sequence of terms. ...

$sum_{n=0}^infty a_n$

is said to converge absolutely if the series of absolute values In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

$sum_{n=0}^infty left|a_nright|$

converges. 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 converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the a_n are real and S is any real number, one can find a reordering so that the reordered series converges with limit S. In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges. ...

Convergence tests

Main article: convergence tests

Comparison test 1: If ∑b_n is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Comparison test 2: If ∑b_n is an absolutely convergent series such that |a_n+1 /a_n | ≤ C |b_n+1 /b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Ratio test: If |a_n+1/a_n| approaches a number less than one as n approaches infinity, then ∑ a_n converges absolutely. When the ratio is 1, convergence can sometimes be determined as well.
Root test: If there exists a constant C < 1 such that |a_n|^1/n ≤ C for all sufficiently large n, then ∑ a_n converges absolutely.
Integral test: if f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = a_n for all n, then ∑ a_n converges if and only if the integral ∫₁^∞ f(x) dx is finite.
Alternating series test: A series of the form ∑ (−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
n-th term test: If lim_n→∞ a _n ≠ 0 then the series diverges.
For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

In mathematics, convergence tests are methods, how to determinate if a series converges or diverges. ... In mathematics, the Comparison test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the Comparison test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ... In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ... In mathematics, the root test is a test for the convergence of an infinite series. ... In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In calculus, the integral of a function is an extension of the concept of a sum. ... The alternating series test is a method used to prove that infinite series of terms converge. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, the nth term test for divergence[1] is a simple test for the divergence of an infinite series: If , then diverges. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... 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. ...

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. For example, the series As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

$sum_{n=0}^inftyfrac{x^n}{n!}$

converges to $e x$ for all x. See also radius of convergence. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or âˆž) such that the series converges if and diverges if In...

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. 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. Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... 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 have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

Dirichlet series

Main article: Dirichlet series

A Dirichlet series is one of the form In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...

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

where s is a complex number. Generally these converge if the real part of s is greater than a number called the abscissa of convergence. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

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 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 and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In mathematics an asymptotic expansion, asymptotic series or PoincarÃ© expansion (after Henri PoincarÃ©) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

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 mathematics, the CesÃ ro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ... In mathematics, a divergent series is an infinite series that does not converge. ... 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. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, 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. Let F be the collection of all finite subsets of I. Note that F is a directed set ordered under inclusion with union as join. We define the sum of the series as the limit In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there... ... In mathematics, inclusion is a partial order on sets. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a join on a set is defined either as unique suprema (least upper bounds) with respect to a partial order on the set, provided such suprema exist, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. ...

$sum_{iin I}a_i = lim_Fleft{sum_{iin A}a_i,bigg|Ain Fright}$

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, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...

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 In mathematics, a well-order (or well-ordering) 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. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

$sum_{iin I}a_i=lim_n left{sum_{i=1}^n a_iright}.$

If this limit exists, then the series converges. 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. A series converges absolutely if In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

$lim_nsum_{i=1}^n |a_i|$

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 topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...

Real sequences

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

$I_n=left{iin I ,bigg | a_i>frac{1}{n}right}$

be the set of indices whose terms are greater than 1/n. Each I_n is finite (otherwise the series would diverge). The set of indices whose terms are nonzero is the union of the I_n by the Archimedean principle, and the union of countably many countable sets is countable by the axiom of choice. In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... 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 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 calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, the counting measure is an intuitive way to put a measure on any set: the size of a subset is taken to be the number of the subsets elements if this is finite, and âˆž if the subset is infinite. ...

The proof goes forward in general first-countable topological vector spaces as well, such as Banach spaces; define I_n to be those indices whose terms are outside the n-th neighborhood of 0. Thus uncountable series can only be interesting if they are valued in spaces that are not first-countable. In topology, 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 local base. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Examples

Given a function f: X→Y, with Y an abelian topological group, then define

$f_a(x)= begin{cases} 0 & xneq a, f(a) & x=a, end{cases}$

the function whose support is a singleton {a}. Then

$f=sum_{x in X}f_a$

in the topology of pointwise convergence. This space is separable but not first countable.
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

$sum_{alphain[0,omega_1]}f(alpha) = omega_1$

(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. This space is not separable.
In the definition of partitions of unity, one constructs sums over arbitrary index. 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 regardly convergence of such sums do not arise.

In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... Generally, a singleton is something which exists alone in some way. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers). ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

References

Bromwich, T.J. An Introduction to the Theory of Infinite Series MacMillan & Co. 1908, revised 1926, reprinted '39, '42, '49, '55, '59, '65.

Categories: Calculus | Mathematical series

Results from FactBites:

Series (mathematics) - Wikipedia, the free encyclopedia (1793 words)
	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 Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
	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.

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Contents

Infinite series GA_googleFillSlot("encyclopedia_square");

Formal definition

History of the theory of infinite series

Development of infinite series

Convergence criteria

Uniform convergence

Semi-convergence

Fourier series

Some types of infinite series

Absolute convergence

Convergence tests

Power series

Dirichlet series

Generalizations

Summations over arbitrary index sets

Real sequences

Examples

See also

References

Infinite series