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# 1 + 2 + 4 + 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two. [1]

## Analytic continuation

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

## 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.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## 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.

## Euler summation

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

## Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

## G. H. Hardy

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

## 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.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## 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").

## 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.

## Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

## Mathematics

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

## Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

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

## Power of two

In mathematics, a power of two is a number of the form where is an integer, i.e. the result of exponentiation with number two as the base and integer as the exponent.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Repeating decimal

A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero.

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## 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.

## Two's complement

Two's complement is a mathematical operation on binary numbers, best known for its role in computing as a method of signed number representation.

## 0.999...

In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).

## 1 + 1 + 1 + 1 + ⋯

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

## 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.

## 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 − 2 + 4 − 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs.

## References

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