Category: Mathematical series

Unconditional convergence
In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent
Formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subt
Series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (additi
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objec
Cauchy–Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a
Convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the fi
Mertens' theorems
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. "Mertens' theorem" may also refer to his theorem in analysis.
Lambert summation
In mathematical analysis, Lambert summation is a summability method for a class of divergent series.
Asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a functio
Weyl sequence
In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl: The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, equidistributed modulo
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 numb
Generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a gener
Liouville–Neumann series
In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.
Kronecker's lemma
In mathematics, Kronecker's lemma (see, e.g., , Lemma IV.3.2)) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proo
Large set (combinatorics)
In combinatorial mathematics, a large set of positive integers is one such that the infinite sum of the reciprocals diverges. A small set is any subset of the positive integers that is not large; that
Neumann series
A Neumann series is a mathematical series of the form where is an operator and its times repeated application. This generalizes the geometric series. The series is named after the mathematician Carl N
Spectrum continuation analysis
Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier
Van Wijngaarden transformation
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Eule
Lidstone series
In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can express certain types of entire functions. Let ƒ(z) be an entire function of exponentia
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henr
Ostrowski–Hadamard gap theorem
In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such
Ramanujan–Sato series
In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, to the form by using other well-defined sequences of integers obeying a certain recurrence relation, sequences whic
Appell series
In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell and that generalize Gauss's hypergeometric series 2F1 of one v
Divergence of the sum of the reciprocals of the primes
The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime number
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 n
Wiener series
In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Vol
General Dirichlet series
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where , are complex numbers and is a strictly increasing sequence of nonnegative real num
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. More precisely, a real or com
Volterra series
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for app
Euler–Boole summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by The concept is named after Leonhard Euler and George Boole. The periodic Euler funct
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
Binomial series
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor
Series multisection
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its
Euler summation
In the mathematics of convergent and divergent series, Euler summation is a summability method. That is, it is a method for assigning a value to a series, different from the conventional method of tak
Madhava of Sangamagrama
Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma (c. 1340 – c. 1425) was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinj
Poincaré series (modular form)
In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several com
In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric f
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. The study of series is a major pa
Alternating series
In mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternatin
Gregory's series
Gregory's series, is an infinite Taylor series expansion of the inverse tangent function.It was discovered in 1668 by James Gregory. It was re-rediscovered a few years later by Gottfried Leibniz, who
Laplace limit
In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately 0.6
Arithmetico-geometric sequence
In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the nth t
Humbert series
In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1F1 of one variable and the
Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resumed formally by expanding the denominator: where the coefficients of the new series are g
Uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function
Bell series
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function
Goldbach–Euler theorem
In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1:
1 − 2 + 3 − 4 + ⋯
In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the
Harmonic progression (mathematics)
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when ea
Lauricella hypergeometric series
In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are: for |x1| + |x2| + |x3| < 1 and for |x1| < 1, |x2| < 1, |x3| < 1 and for |x1|½ +
Kolmogorov's three-series theorem
In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the con
Fabry gap theorem
In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series
Conditional convergence
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Edgeworth series
The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability d
Kempner series
The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum where the prime indicates
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 18
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series is a Puiseux series in the indetermin
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formula
Mercator series
In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: In summation notation, The series converges to the natural logarithm (shifted by 1) whenev
Madhava series
In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by an Indian Mathematician
List of mathematical series
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. * Here, is taken to have the value * denotes the
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. If a series converge
Abelian and Tauberian theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The
Hahn series
In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of form
Telescoping series
In mathematics, a telescoping series is a series whose general term can be written as , i.e. the difference of two consecutive terms of a sequence . As a consequence the partial sums only consists of
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modu
Charles Haros
Charles Haros was a geometer (mathematician) in the French at the end of the eighteenth century and the beginning of the nineteenth century.
Problems involving arithmetic progressions
Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view.
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel. It is particularly useful for summing divergent asymptotic series, and in some sense gives the be
Function series
In calculus, a function series is a series, where the summands are not just real or complex numbers but functions.
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
Sturm series
In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-
Sequence transformation
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequ
Lévy–Steinitz theorem
In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published pa
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. Power series are useful in mathematical anal
Rational series
In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a se
In mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions. Epi-convergence is important because it is the appropriate notion of convergence
Method of dominant balance
In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterati
Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of importa
Leibniz formula for π
In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general seri