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# 1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is the infinite series whose terms are the successive positive integers, given alternating signs. [1]

56 relations: Absolute value, Alternating series, Apéry's constant, Arithmetic mean, Émile Borel, Basel problem, Bernoulli number, Borel summation, Calculus, Cauchy product, Cesàro summation, Convolution, Countable set, Derivative, Dirichlet eta function, Dirichlet series, Divergent series, Empty sum, Ernesto Cesàro, Eugène Charles Catalan, Euler summation, Finite difference, French language, Function (mathematics), Functional equation, Generalization, Geometric series, Grandi's series, Integer, Iterated function, John C. Baez, Leonhard Euler, Limit of a sequence, Mathematics, Mean value theorem, Natural number, Niels Henrik Abel, Otto Hölder, Parity (mathematics), Physics, Pointed set, Polynomial long division, Quantum harmonic oscillator, Riemann integral, Riemann zeta function, Series (mathematics), Square (algebra), Summation, Taylor series, Taylor's theorem, ... Expand index (6 more) »

## Absolute value

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

## Alternating series

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

## Apéry's constant

In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number where is the Riemann zeta 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.

## Émile Borel

Félix Édouard Justin Émile Borel (7 January 1871 – 3 February 1956) was a French mathematician and politician.

## 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''.

## Bernoulli number

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

## Borel summation

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

## Calculus

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.

## Cauchy product

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

## 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.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Derivative

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).

## Dirichlet eta function

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, &zeta;(s) &mdash; and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted &zeta;*(s).

## 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 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.

## Empty sum

In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.

## Ernesto Cesàro

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

## Eugène Charles Catalan

Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics.

## Euler summation

In the mathematics of convergent and divergent series, Euler summation is a summability method.

## Finite difference

A finite difference is a mathematical expression of the form.

## French language

French (le français or la langue française) is a Romance language of the Indo-European family.

## Function (mathematics)

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

## Functional equation

In mathematics, a functional equation is any equation in which the unknown represents a function.

## Generalization

A generalization (or generalisation) is the formulation of general concepts from specific instances by abstracting common properties.

## Geometric series

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

## 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.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Iterated function

In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times.

## John C. Baez

John Carlos Baez (born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California.

## 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.

## Mathematics

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

## Mean value theorem

In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

## 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").

## 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.

## Otto Hölder

Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.

## Parity (mathematics)

In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.

## 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.

## Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.

## Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division.

## Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.

## Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

## 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.

## 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.

## Square (algebra)

In mathematics, a square is the result of multiplying a number by itself.

## Summation

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

## 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.

## Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

## Term test

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

## Tetrahedral number

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron.

## Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right.

## Well-defined

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## 1 + 2 + 3 + 4 + ⋯

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

## References

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