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# 1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + · · · is a divergent series. [1]

58 relations: A Disappearing Number, Additive identity, Alternating series, Analytic continuation, Asymptote, Bernoulli number, Bosonic string theory, Bounded function, Brady Haran, Cesàro summation, Complex analysis, David Leavitt, Dirichlet eta function, Dirichlet series, Divergent series, Edward Frenkel, Euler–Maclaurin formula, G. H. Hardy, Goddard–Thorn theorem, Grandi's series, Infinity, John C. Baez, John Edensor Littlewood, Limit of a sequence, Luboš Motl, Mollifier, Monstrous moonshine, Morris Kline, Natural number, One-sided limit, Pythagoreanism, Quantum field theory, Quantum harmonic oscillator, Ramanujan summation, Real analysis, Riemann zeta function, Scalar field, Scientific American, Sequence, Series (mathematics), Simon McBurney, Smoothness, Srinivasa Ramanujan, String theory, Support (mathematics), Terence Tao, Term test, The Indian Clerk, The New York Times, Thomas John I'Anson Bromwich, ... Expand index (8 more) »

## A Disappearing Number

A Disappearing Number is a 2007 play co-written and devised by the Théâtre de Complicité company and directed and conceived by English playwright Simon McBurney.

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

## Alternating series

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

## Analytic continuation

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

## Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

## Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory.

## Bosonic string theory

Bosonic string theory is the original version of string theory, developed in the late 1960s.

## Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.

Brady John Haran (born 18 June 1976) is an Australian independent film-maker and video journalist who is known for his educational videos and documentary films produced for BBC News and for his YouTube channels, such as Numberphile and Periodic Videos.

## Cesàro summation

In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not convergent in the usual sense, while coinciding with the standard sum if they are convergent.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## David Leavitt

David Leavitt (born June 23, 1961) is an American writer of novels, short stories, and non-fiction.

## 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, ζ(s) &mdash; and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s).

## Dirichlet series

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

## Edward Frenkel

Edward Vladimirovich Frenkel is a mathematician working in representation theory, algebraic geometry, and mathematical physics.

## Euler–Maclaurin formula

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

## G. H. Hardy

Godfrey Harold ("G. H.") Hardy FRS (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

## Goddard–Thorn theorem

In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector spaces.

## Grandi's series

In mathematics, the infinite series 1 − 1 + 1 − 1 +..., 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.

## Infinity

Infinity (symbol) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics.

## 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.

## John Edensor Littlewood

John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician, best known for his achievements in analysis, number theory and differential equations.

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n sin(1/n) becomes arbitrarily close to 1.

## Luboš Motl

Luboš Motl (born December 5, 1973) is a Czech theoretical physicist by training who was an assistant professor at Harvard University from 2004 to 2007.

## Mollifier

In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

## Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the ''j'' function.

## Morris Kline

Morris Kline (May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.

## Natural number

In mathematics, the natural numbers (sometimes called the whole numbers): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number.

## One-sided limit

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above.

## Pythagoreanism

Pythagoreanism originated in the 5th century BCE, based on teachings, or beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics, music and astronomy.

## Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

## Quantum harmonic oscillator

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

## Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series.

## Real analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.

## Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space.

## Scientific American

Scientific American (informally abbreviated SciAm) is an American popular science magazine.

## Sequence

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

## Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence.

## Simon McBurney

Simon Montagu McBurney, OBE (born 25 August 1957) is an English actor, writer and director.

## Smoothness

In mathematical analysis, smoothness has to do with how many derivatives of a function exist and are continuous.

## Srinivasa Ramanujan

Srinivasa Ramanujan Iyengar (22 December 188726 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.

## String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

## Support (mathematics)

In mathematics, the support of a function is the set of points where the function is not zero-valued or, in the case of functions defined on a topological space, the closure of that set.

## Terence Tao

Terence "Terry" Chi-Shen Tao FAA FRS (born 17 July 1975, Adelaide), is an Australian mathematician who has worked in various areas of mathematics.

## Term test

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

## The Indian Clerk

The Indian Clerk is a fictive biographical novel by David Leavitt, published in 2007.

## The New York Times

The New York Times (NYT) is an American daily newspaper, founded and continuously published in New York City since September 18, 1851, by the New York Times Company.

## Thomas John I'Anson Bromwich

Thomas John I'Anson Bromwich (1875–1929) was an English mathematician, and a Fellow of the Royal Society.

## Transverse wave

A transverse wave is a moving wave that consists of oscillations occurring perpendicular (or right angled) to the direction of energy transfer.

## Triangular number

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

## University of Nottingham

The University of Nottingham is a public research university based in Nottingham, Nottinghamshire, England, United Kingdom.

YouTube is a video-sharing website headquartered in San Bruno, California, United States.

## 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.

## 1 + 1 + 1 + 1 + ⋯

In mathematics, 1 + 1 + 1 + 1 + · · ·, also written \sum_^ n^0, \sum_^ 1^n, or simply \sum_^ 1, is a divergent series, meaning that its sequence of partial sums do not converge to a limit in the real numbers.

## 1 + 2 + 4 + 8 + ⋯

In mathematics, 1 + 2 + 4 + 8 + … is the infinite series whose terms are the successive powers of two.

## 1 − 2 + 3 − 4 + ⋯

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

## References

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