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Series (mathematics)

Index Series (mathematics)

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. [1]

200 relations: Abel's test, Abelian group, Absolute convergence, Absolute value, Actual infinity, Addition, Alfred Pringsheim, Almost everywhere, Alternating series, Alternating series test, Analytic continuation, Analytic function, Analytic number theory, Angelo Genocchi, Annulus (mathematics), Antiderivative, Archimedes, Area, Arithmetic progression, Arithmetico–geometric sequence, Arndt, Arthur Cayley, Associative algebra, Asymptotic expansion, Augustin-Louis Cauchy, Augustus De Morgan, Banach limit, Banach space, Basel problem, Binomial series, Borel summation, Bounded set, Brook Taylor, Calculus, Carl Friedrich Gauss, Carl Johan Malmsten, Cauchy condensation test, Cauchy product, Cauchy sequence, Cesàro summation, Charles Hermite, Closed set, Colin Maclaurin, Combinatorics, Commutative ring, Compact convergence, Compact space, Complete metric space, Complex number, Computer science, ..., Conditional convergence, Continued fraction, Convergence of Fourier series, Convergence tests, Convergent series, Counting measure, Crelle's Journal, Decimal, Derivative, Differential algebra, Dini test, Direct comparison test, Directed set, Dirichlet series, Divergent series, Ernst Kummer, Expression (mathematics), Faulhaber's formula, Field (mathematics), Finance, Finite difference, Finite set, First uncountable ordinal, First-countable space, Formal sum, Fourier series, François Viète, Function (mathematics), General Dirichlet series, Generating function, Geometric series, Georges Henri Halphen, Gotthold Eisenstein, Graded ring, Greek mathematics, Harmonic series (mathematics), Hausdorff space, Hilbert–Poincaré series, Hypergeometric function, If and only if, Inclusion map, Index set, Infinite compositions of analytic functions, Infinite expression, Infinite product, Integral, Integral test for convergence, Integration by parts, Interval (mathematics), Iterated binary operation, Jacob Bernoulli, James Gregory (mathematician), James Whitbread Lee Glaisher, Józef Maria Hoene-Wroński, Johann Bernoulli, Join and meet, Joseph Bertrand, Joseph Fourier, Joseph Ludwig Raabe, Joseph-Louis Lagrange, Karl Schröter, Karl Weierstrass, Leonhard Euler, Limit (mathematics), Limit comparison test, Limit of a sequence, Linear function, Linear map, Liouville number, List of mathematical series, Louis Poinsot, Ludwig Schläfli, Mathematical analysis, Mathematical constant, Mathematics, Measure (mathematics), Method of exhaustion, Metric space, Modes of convergence, Monoid, Monotonic function, Multiplication, Natural number, Net (mathematics), Nicolas Bourbaki, Niels Henrik Abel, Null set, Number, Order topology, Ordinal number, Oscar Schlömilch, Pafnuty Chebyshev, Parabola, Paradox, Partially ordered set, Partition of unity, Pauker, Paul Émile Appell, Paul du Bois-Reymond, Philipp Ludwig von Seidel, Philosopher, Physics, Pi, Pierre Ossian Bonnet, Pointwise convergence, Power series, Prefix sum, Q-Pochhammer symbol, Radius of convergence, Ratio test, Real number, Repeating decimal, Riemann series theorem, Riemann zeta function, Ring (mathematics), Root test, Rudolf Lipschitz, Sequence, Sequence transformation, Series expansion, Silverman–Toeplitz theorem, Siméon Denis Poisson, Singleton (mathematics), Sir George Stokes, 1st Baronet, Statistics, Subset, Summation, Summation by parts, Support (mathematics), Taylor series, Telescoping series, Term (logic), Term test, Thomas John I'Anson Bromwich, Topological group, Total algebra, Total order, Transfinite induction, Trigonometric functions, Ulisse Dini, Unary operation, Uniform convergence, Union (set theory), University of St Andrews, Weierstrass M-test, Well-order, Zeno of Elea, Zeno's paradoxes, Zeros and poles, 0.999.... Expand index (150 more) »

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.

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Abelian group

In abstract algebra, 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.

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Absolute convergence

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.

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Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Actual infinity

In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects.

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Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

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Alfred Pringsheim

Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts.

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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.

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Alternating series

In mathematics, an alternating series is an infinite series of the form with an > 0 for all n.

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Alternating series test

In mathematical analysis, the alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series.

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Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

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Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

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Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.

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Angelo Genocchi

Angelo Genocchi (5 March 1817 – 7 March 1889) was an Italian mathematician who specialized in number theory.

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Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring" is anulus/annulus, with plural anuli/annuli) is a ring-shaped object, a region bounded by two concentric circles.

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In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

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Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

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Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

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Arithmetico–geometric sequence

In mathematics, an arithmetico–geometric sequence is the result of the term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression.

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Arndt or Arnd is a German masculine given name, a short form of Arnold, as well as a German patronymic surname.

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Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

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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 a given function as the argument of the function tends towards a particular, often infinite, point.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (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.

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Augustus De Morgan

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician.

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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.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''.

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Binomial series

In mathematics, the binomial series is the Maclaurin series for the function f given by f(x).

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Borel summation

In mathematics, Borel summation is a summation method for divergent series, introduced by.

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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Brook Taylor

Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series.

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Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), 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 arithmetic operations.

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Carl Johan Malmsten

Carl Johan Malmsten (April 9, 1814 in Uddetorp, Skara County, Sweden – February 11, 1886 in Uppsala, Sweden) was a Swedish mathematician and politician.

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Cauchy condensation test

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series.

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Cauchy product

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

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Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

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Cesàro summation

In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not convergent in the usual sense.

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Charles Hermite

Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Colin Maclaurin

Colin Maclaurin (Cailean MacLabhruinn; 1 February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra.

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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

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Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

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Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

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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 a field known as classical harmonic analysis, a branch of pure mathematics.

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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.

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Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

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Counting measure

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 elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.

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Crelle's Journal

Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).

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The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers.

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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).

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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.

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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.

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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 or an improper integral.

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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 ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.

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Dirichlet series

In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.

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Divergent series

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.

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Ernst Kummer

Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.

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Expression (mathematics)

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

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Faulhaber's formula

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers Bj, in the form submitted by Jacob Bernoulli and published in 1713: where p^\underline.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Finance is a field that is concerned with the allocation (investment) of assets and liabilities (known as elements of the balance statement) over space and time, often under conditions of risk or uncertainty.

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Finite difference

A finite difference is a mathematical expression of the form.

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Finite set

In mathematics, a finite set is a set that has a finite number of elements.

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First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable.

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First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".

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Formal sum

In mathematics, a formal sum, formal series, or formal linear combination may be.

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Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

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François Viète

François Viète (Franciscus Vieta; 1540 – 23 February 1603), Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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General Dirichlet series

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends to infinity.

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Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

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Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms.

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Georges Henri Halphen

Georges Henri Halphen (30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician.

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Gotthold Eisenstein

Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician.

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Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_.

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Greek mathematics

Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.

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Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are,,, etc., of the string's fundamental wavelength.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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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 graded algebraic structures (where the dimension of the entire structure is often infinite).

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Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Inclusion map

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.

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Index set

In mathematics, an index set is a set whose members label (or index) members of another set.

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Infinite compositions of analytic functions

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions.

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Infinite expression

In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth.

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Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3,...

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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence.

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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 their derivative and antiderivative.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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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.

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Jacob Bernoulli

Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.

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James Gregory (mathematician)

James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer.

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James Whitbread Lee Glaisher

James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher the meteorologist and Cecilia Glaisher the photographer, was a prolific English mathematician and astronomer.

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Józef Maria Hoene-Wroński

Józef Maria Hoene-Wroński (Josef Hoëné-Wronski,; 23 August 1776 – 9 August 1853) was a Polish Messianist philosopher, mathematician, physicist, inventor, lawyer, and economist.

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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.

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Join and meet

In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S.

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Joseph Bertrand

Joseph Louis François Bertrand (11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics.

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Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

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Joseph Ludwig Raabe

Joseph Ludwig Raabe (15 May 1801 in Brody, Galicia – 22 January 1859 in Zürich, Switzerland) was a Swiss mathematician.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

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Karl Schröter

Karl Walter Schröter (* 7 September 1905 in Biebrich near Wiesbaden, † 22 August 1977 in Berlin) was a German mathematician and logician.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

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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.

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Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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Linear function

In mathematics, the term linear function refers to two distinct but related notions.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Liouville number

In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers.

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List of mathematical series

This list of mathematical series contains formulae for finite and infinite sums.

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Louis Poinsot

Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist.

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Ludwig Schläfli

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical constant

A mathematical constant is a special number that is "significantly interesting in some way".

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

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Method of exhaustion

The method of exhaustion (methodus exhaustionibus, or méthode des anciens) 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.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Modes of convergence

In mathematics, there are many senses in which a sequence or a series is said to be convergent.

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

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Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

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Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

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Natural number

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

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Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.

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Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

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Niels Henrik Abel

Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.

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Null set

In set theory, a null set N \subset \mathbb is a set that can be covered by a countable union of intervals of arbitrarily small total length.

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A number is a mathematical object used to count, measure and also label.

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Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.

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Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

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Oscar Schlömilch

Oscar (Oskar) Xavier Schlömilch (13 April 1823 – 7 February 1901) was a German mathematician, born in Weimar, working in mathematical analysis.

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Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician.

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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.

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Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

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Partition of unity

In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unit interval such that for every point, x\in X,.

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Pauker or Paucker (Паукер, poyker) is a surname of German origin.

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Paul Émile Appell

Paul Appell (27 September 1855 in Strasbourg – 24 October 1930 in Paris), also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris.

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Paul du Bois-Reymond

Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German mathematician who was born in Berlin and died in Freiburg.

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Philipp Ludwig von Seidel

Philipp Ludwig von Seidel (23 October 1821 in Zweibrücken, Germany – 13 August 1896 in Munich, German Empire) was a German mathematician.

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A philosopher is someone who practices philosophy, which involves rational inquiry into areas that are outside either theology or science.

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Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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The number is a mathematical constant.

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Pierre Ossian Bonnet

Pierre Ossian Bonnet (22 December 1819, Montpellier – 22 June 1892, Paris) was a French mathematician.

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Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.

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Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

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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 of the natural numbers are the triangular numbers: |- !input numbers | 1 || 2 || 3 || 4 || 5 || 6 ||...

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Q-Pochhammer symbol

In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a ''q''-analog of the Pochhammer symbol.

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Radius of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

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Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and is nonzero when is large.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Repeating decimal

A repeating 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.

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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 can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges.

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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Root test

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.

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Rudolf Lipschitz

Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as number theory, algebras with involution and classical mechanics.

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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Sequence transformation

In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space).

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Series expansion

In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).

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Silverman–Toeplitz theorem

In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular.

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Siméon Denis Poisson

Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.

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Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

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Sir George Stokes, 1st Baronet

Sir George Gabriel Stokes, 1st Baronet, (13 August 1819 – 1 February 1903), was an Irish physicist and mathematician.

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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

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In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

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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.

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Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

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Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation.

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Term (logic)

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.

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Term test

In mathematics, the nth-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series.

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Thomas John I'Anson Bromwich

Thomas John I'Anson Bromwich (1875–1929) was an English mathematician, and a Fellow of the Royal Society.

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Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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Total algebra

In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring.

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Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

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Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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Ulisse Dini

Ulisse Dini (14 November 1845 – 28 October 1918) was an Italian mathematician and politician, born in Pisa.

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Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.

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Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.

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Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

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University of St Andrews

The University of St Andrews (informally known as St Andrews University or simply St Andrews; abbreviated as St And, from the Latin Sancti Andreae, in post-nominals) is a British public research university in St Andrews, Fife, Scotland.

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Weierstrass M-test

In mathematics, the Weierstrass M-test is a test for testing whether an infinite series of functions converges uniformly and absolutely.

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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.

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Zeno of Elea

Zeno of Elea (Ζήνων ὁ Ἐλεάτης) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides.

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Zeno's paradoxes

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 plurality and change is mistaken, and in particular that motion is nothing but an illusion.

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Zeros and poles

In mathematics, a zero of a function is a value such that.

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In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).

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[1] https://en.wikipedia.org/wiki/Series_(mathematics)

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