## Table of Contents

60 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, French language, Function (mathematics), Functional equation, Generalization, Generating function, 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, Nth-term test, Otto Hölder, Parity (mathematics), Physics, Pointed set, Polynomial long division, Quantum harmonic oscillator, Recurrence relation, Riemann integral, Riemann zeta function, Series (mathematics), Square (algebra), Summation, ... Expand index (10 more) »

- Divergent series
- Mathematical paradoxes

## Absolute value

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

See 1 − 2 + 3 − 4 + ⋯ and Absolute value

## Alternating series

In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all. 1 − 2 + 3 − 4 + ⋯ and alternating series are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Alternating series

## Apéry's constant

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes.

See 1 − 2 + 3 − 4 + ⋯ and Apéry's constant

## Arithmetic mean

In mathematics and statistics, the arithmetic mean, arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection.

See 1 − 2 + 3 − 4 + ⋯ and Arithmetic mean

## Émile Borel

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

See 1 − 2 + 3 − 4 + ⋯ and Émile Borel

## Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.

See 1 − 2 + 3 − 4 + ⋯ and Basel problem

## Bernoulli number

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

See 1 − 2 + 3 − 4 + ⋯ and Bernoulli number

## Borel summation

In mathematics, Borel summation is a summation method for divergent series, introduced by. 1 − 2 + 3 − 4 + ⋯ and Borel summation are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Borel summation

## Calculus

Calculus 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.

See 1 − 2 + 3 − 4 + ⋯ and Calculus

## Cauchy product

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

See 1 − 2 + 3 − 4 + ⋯ and Cauchy product

## Cesàro summation

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

See 1 − 2 + 3 − 4 + ⋯ and Cesàro summation

## Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g).

See 1 − 2 + 3 − 4 + ⋯ and Convolution

## Countable set

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.

See 1 − 2 + 3 − 4 + ⋯ and Countable set

## Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input.

See 1 − 2 + 3 − 4 + ⋯ and Derivative

## 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: \eta(s).

See 1 − 2 + 3 − 4 + ⋯ and Dirichlet eta function

## Dirichlet series

In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where s is complex, and a_n is a complex sequence. 1 − 2 + 3 − 4 + ⋯ and Dirichlet series are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Dirichlet series

## 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 − 2 + 3 − 4 + ⋯ and divergent series are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Divergent series

## Empty sum

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

See 1 − 2 + 3 − 4 + ⋯ and Empty sum

## Ernesto Cesàro

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

See 1 − 2 + 3 − 4 + ⋯ and Ernesto Cesàro

## 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.

See 1 − 2 + 3 − 4 + ⋯ and Eugène Charles Catalan

## Euler summation

In the mathematics of convergent and divergent series, Euler summation is a summation method. 1 − 2 + 3 − 4 + ⋯ and Euler summation are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Euler summation

## French language

French (français,, or langue française,, or by some speakers) is a Romance language of the Indo-European family.

See 1 − 2 + 3 − 4 + ⋯ and French language

## Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See 1 − 2 + 3 − 4 + ⋯ and Function (mathematics)

## Functional equation

In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns.

See 1 − 2 + 3 − 4 + ⋯ and Functional equation

## Generalization

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims.

See 1 − 2 + 3 − 4 + ⋯ and Generalization

## Generating function

In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.

See 1 − 2 + 3 − 4 + ⋯ and Generating function

## Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. 1 − 2 + 3 − 4 + ⋯ and geometric series are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Geometric series

## Grandi's series

In mathematics, the infinite series, also written is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. 1 − 2 + 3 − 4 + ⋯ and Grandi's series are divergent series, mathematical paradoxes and mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Grandi's series

## Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See 1 − 2 + 3 − 4 + ⋯ and Integer

## Iterated function

In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times.

See 1 − 2 + 3 − 4 + ⋯ and Iterated function

## 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.

See 1 − 2 + 3 − 4 + ⋯ and John C. Baez

## Leonhard Euler

Leonhard Euler (15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

See 1 − 2 + 3 − 4 + ⋯ and Leonhard Euler

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1.

See 1 − 2 + 3 − 4 + ⋯ and Limit of a sequence

## Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See 1 − 2 + 3 − 4 + ⋯ and Mathematics

## Mean value theorem

In mathematics, the mean value theorem (or Lagrange 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.

See 1 − 2 + 3 − 4 + ⋯ and Mean value theorem

## Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

See 1 − 2 + 3 − 4 + ⋯ and Natural number

## 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.

See 1 − 2 + 3 − 4 + ⋯ and Niels Henrik Abel

## Nth-term test

In mathematics, the nth-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:If \lim_ a_n \neq 0 or if the limit does not exist, then \sum_^\infty a_n diverges.

See 1 − 2 + 3 − 4 + ⋯ and Nth-term test

## Otto Hölder

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

See 1 − 2 + 3 − 4 + ⋯ and Otto Hölder

## Parity (mathematics)

In mathematics, parity is the property of an integer of whether it is even or odd.

See 1 − 2 + 3 − 4 + ⋯ and Parity (mathematics)

## Physics

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

See 1 − 2 + 3 − 4 + ⋯ and Physics

## 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.

See 1 − 2 + 3 − 4 + ⋯ and Pointed set

## 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 generalized version of the familiar arithmetic technique called long division.

See 1 − 2 + 3 − 4 + ⋯ and Polynomial long division

## Quantum harmonic oscillator

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

See 1 − 2 + 3 − 4 + ⋯ and Quantum harmonic oscillator

## Recurrence relation

In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms.

See 1 − 2 + 3 − 4 + ⋯ and Recurrence relation

## 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.

See 1 − 2 + 3 − 4 + ⋯ and Riemann integral

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s).

See 1 − 2 + 3 − 4 + ⋯ and Riemann zeta function

## Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. 1 − 2 + 3 − 4 + ⋯ and series (mathematics) are mathematical series.

See 1 − 2 + 3 − 4 + ⋯ and Series (mathematics)

## Square (algebra)

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

See 1 − 2 + 3 − 4 + ⋯ and Square (algebra)

## Summation

In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.

See 1 − 2 + 3 − 4 + ⋯ and Summation

## Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.

See 1 − 2 + 3 − 4 + ⋯ and Taylor series

## Taylor's theorem

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

See 1 − 2 + 3 − 4 + ⋯ and Taylor's theorem

## 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.

See 1 − 2 + 3 − 4 + ⋯ and Tetrahedral number

## Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle.

See 1 − 2 + 3 − 4 + ⋯ and Triangular number

## Well-defined expression

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

See 1 − 2 + 3 − 4 + ⋯ and Well-defined expression

## 0

0 (zero) is a number representing an empty quantity.

## 1 + 1 + 1 + 1 + ⋯

In mathematics,, also written,, or simply, is a divergent series. 1 − 2 + 3 − 4 + ⋯ and 1 + 1 + 1 + 1 + ⋯ are divergent series and mathematical paradoxes.

See 1 − 2 + 3 − 4 + ⋯ and 1 + 1 + 1 + 1 + ⋯

## 1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series. 1 − 2 + 3 − 4 + ⋯ and 1 + 2 + 3 + 4 + ⋯ are divergent series.

See 1 − 2 + 3 − 4 + ⋯ and 1 + 2 + 3 + 4 + ⋯

## 1 + 2 + 4 + 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two. 1 − 2 + 3 − 4 + ⋯ and 1 + 2 + 4 + 8 + ⋯ are divergent series.

See 1 − 2 + 3 − 4 + ⋯ and 1 + 2 + 4 + 8 + ⋯

## 1 − 2 + 4 − 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. 1 − 2 + 3 − 4 + ⋯ and 1 − 2 + 4 − 8 + ⋯ are divergent series.

See 1 − 2 + 3 − 4 + ⋯ and 1 − 2 + 4 − 8 + ⋯

## See also

### Divergent series

- 1 − 1 + 2 − 6 + 24 − 120 + ⋯
- 1 − 2 + 3 − 4 + ⋯
- 1 − 2 + 4 − 8 + ⋯
- 1 + 1 + 1 + 1 + ⋯
- 1 + 2 + 3 + 4 + ⋯
- 1 + 2 + 4 + 8 + ⋯
- Antilimit
- Divergent geometric series
- Divergent series
- Grandi's series
- Harmonic series (mathematics)
- History of Grandi's series
- Occurrences of Grandi's series
- Summability methods
- Summation of Grandi's series

### Mathematical paradoxes

- 0.999...
- 1 − 2 + 3 − 4 + ⋯
- 1 + 1 + 1 + 1 + ⋯
- All horses are the same color
- Banach–Tarski paradox
- Berry paradox
- Bertrand paradox (probability)
- Braess's paradox
- Chessboard paradox
- Coin rotation paradox
- Cramer's paradox
- Curry's paradox
- Gabriel's horn
- Grandi's series
- Hausdorff paradox
- Hilbert's paradox of the Grand Hotel
- Hilbert–Bernays paradox
- Hooper's paradox
- Interesting number paradox
- Kleene–Rosser paradox
- Knower paradox
- Missing square puzzle
- Newcomb's paradox
- Paradoxes of infinity
- Paradoxes of set theory
- Parrondo's paradox
- Potato paradox
- Richard's paradox
- Schwarz lantern
- Skolem's paradox
- Sphere eversion
- Staircase paradox
- String girdling Earth
- The Banach–Tarski Paradox (book)
- Vanishing puzzle
- Von Neumann paradox
- Zeno's paradoxes

## References

Also known as 1 - 2 + 3 - 4, 1 - 2 + 3 - 4 +, 1 - 2 + 3 - 4 + . . ., 1 - 2 + 3 - 4 + ..., 1 - 2 + 3 - 4 + · · ·, 1 - 2 + 3 - 4 ..., 1 − 2 + 3, 1 − 2 + 3 − 4 + · ·, 1 − 2 + 3 − 4 + ···, 1 − 2 3 − 4 · · ·, 1- 2 + 3 - 4, 1-2+3-4, 1-2+3-4+, 1-2+3-4+..., 1-2+3-4+···.