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# 1 + 2 + 4 + 8 + ⋯

In mathematics, 1 + 2 + 4 + 8 + … 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 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 plane

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

## 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 ("G. H.") Hardy FRS (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 +..., 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.

## 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 (17071783) was a pioneering Swiss mathematician and physicist.

## Mathematical proof

In mathematics, a proof is a deductive argument for a mathematical statement.

## Mathematics

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

## Möbius transformation

In geometry and complex analysis, a Möbius transformation of the 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 way different from the extension of the rational number system to the real and complex number systems.

## Power of two

In mathematics, a power of two means 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, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c).

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 that represents a quantity along a continuous line.

## Repeating decimal

A repeating or recurring decimal is a way of representing rational numbers in base 10 arithmetic.

## Riemann sphere

In mathematics, the Riemann sphere, named after the 19th century mathematician Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Series (mathematics)

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

## Two's complement

Two's complement is a mathematical operation on binary numbers, as well as a binary signed number representation based on this operation.

## 0.999...

In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, for example as 0.9..., or in a variety of other variants such as 0.9, 0.(9), or) denotes a real number that can be shown to be the number ''one''.

## 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.

## References

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