Series (mathematics)

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Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellanea Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
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In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other.^[1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes.^[2][3] Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola.^[4][5] The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton.^[6] The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy,^[7] among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

In modern terminology, any ordered infinite sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ of terms, whether those terms are numbers, functions, matrices, or anything else that can be added, defines a series, which is the addition of the a_i one after the other. To emphasize that there are an infinite number of terms, series are often also called infinite series. Series are represented by an expression like $${\displaystyle a_{1}+a_{2}+a_{3}+\cdots ,}$$ or, using capital-sigma summation notation,^[8] $${\displaystyle \sum _{i=1}^{\infty }a_{i}.}$$

The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity of the finite sums of the n first terms of the series if the limit exists.^[9][10][11] These finite sums are called the partial sums of the series. Using summation notation, $${\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{n\to \infty }\,\sum _{i=1}^{n}a_{i},}$$ if it exists.^[9][10][11] When the limit exists, the series is convergent or summable and also the sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ is summable, and otherwise, when the limit does not exist, the series is divergent.^[9][10][11]

The expression ${\textstyle \sum _{i=1}^{\infty }a_{i}}$ denotes both the series—the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting by ${\displaystyle a+b}$ both the addition—the process of adding—and its result—the sum of a and b.

Commonly, the terms of a series come from a ring, often the field ${\displaystyle \mathbb {R} }$ of the real numbers or the field ${\displaystyle \mathbb {C} }$ of the complex numbers. If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is the Cauchy product.^[12][13][14]