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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. [1]

58 relations: Abel's theorem, Abelian and tauberian theorems, Albert Ingham, Alfred Tauber, Analytic continuation, Arithmetic mean, Augustin-Louis Cauchy, Average, Axiom of choice, Banach algebra, Banach limit, Bernoulli number, Bijection, Borel summation, Cesàro summation, Consistency, Convergent series, Dirichlet series, Divergent geometric series, Divergent series, Ernesto Cesàro, Euler–Maclaurin formula, Extrapolation, Ferdinand Georg Frobenius, Fourier analysis, Grandi's series, Hahn–Banach theorem, Harmonic series (mathematics), Henri Poincaré, Lambert summation, Leonhard Euler, Limit of a sequence, Mathematical analysis, Mathematics, Mittag-Leffler star, Nicole Oresme, Niels Erik Nørlund, Niels Henrik Abel, Padé approximant, Partial function, Perturbation theory, Physics, Positive real numbers, Quantum mechanics, Regularization (physics), Renormalization, Riemann zeta function, Sequence, Sequence transformation, Series (mathematics), ... Expand index (8 more) »

## Abel's theorem

In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients.

## 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.

## Albert Ingham

Albert Edward Ingham FRS (3 April 1900 – 6 September 1967) was an English mathematician.

## Alfred Tauber

Alfred Tauber (5 November 1866 – 26 July 1942) was an Austrian and Slovak mathematician, known for his contribution to mathematical analysis and to the theory of functions of a complex variable: he is the eponym of an important class of theorems with applications ranging from mathematical and harmonic analysis to number theory.

## Analytic continuation

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

## Arithmetic mean

In mathematics and statistics, the arithmetic mean (stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.

## 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.

## Average

In colloquial language, an average is a middle or typical number of a list of numbers.

## Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

## Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.

## 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.

## Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Borel summation

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

## 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.

## Consistency

In classical deductive logic, a consistent theory is one that does not contain a contradiction.

## Convergent series

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

## Dirichlet series

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

## Divergent geometric series

In mathematics, an infinite geometric series of the form is divergent if and only if | r | ≥ 1.

## 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.

## Ernesto Cesàro

Ernesto Cesàro (12 March 1859 &ndash; 12 September 1906) was an Italian mathematician who worked in the field of differential geometry.

## Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums.

## Extrapolation

In mathematics, extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable.

## Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

## Grandi's series

In mathematics, the infinite series 1 - 1 + 1 - 1 + \dotsb, also written \sum_^ (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703.

## Hahn–Banach theorem

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.

## 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.

## Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

## Lambert summation

In mathematical analysis, Lambert summation is a summability method for a class of divergent series.

## 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.

## 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.

## 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.

## Mathematics

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

## Mittag-Leffler star

In complex analysis, a branch of mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the complex plane obtained by attempting to extend that function along rays emanating from a given point.

## Nicole Oresme

Nicole Oresme (c. 1320–1325 – July 11, 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a significant philosopher of the later Middle Ages.

## Niels Erik Nørlund

Niels Erik Nørlund (26 October 1885, in Slagelse &ndash; 4 July 1981, in Copenhagen) was a Danish mathematician.

## 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.

In mathematics a Padé approximant is the 'best' approximation of a function by a rational function of given order – under this technique, the approximant's power series agrees with the power series of the function it is approximating.

## Partial function

In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.

## Perturbation theory

Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

## Physics

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.

## Positive real numbers

In mathematics, the set of positive real numbers, \mathbb_.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Regularization (physics)

In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called regulator.

## Renormalization

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions.

## 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.

## Sequence

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

## Sequence transformation

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

## 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.

## Shift rule

The shift rule is a mathematical rule for sequences and series.

## 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.

## 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.

## Wiener's tauberian theorem

In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.

## Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.

## Zorn's lemma

Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.

## 1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series.

## 1 − 1 + 2 − 6 + 24 − 120 + ...

In mathematics, the divergent series was first considered by Euler, who applied summability methods to assign a finite value to the series.

## References

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