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# 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]

## Analytic continuation

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

## 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.

## Emilio Elizalde

Emilio Elizalde (born March 8, 1950) is a Spanish physicist working in the fields of gravitational physics and general relativity.

## Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively).

## Francisco José Ynduráin

Francisco José Ynduráin Muñoz (23 December 1940 – 6 June 2008) was a Spanish theoretical physicist.

## Geometric series

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

## 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 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength.

## Limit of a sequence

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

## Mathematics

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

## Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order.

In mathematics the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator q not equal to zero.

## Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

## Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.

## 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.

## Series (mathematics)

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

## Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena.

## 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 + 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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