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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 definition of a given analytic function.

## 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.

## Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers.

## 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 summation method.

## Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation.

## 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 the sum of an infinite number of terms that have a constant ratio between successive terms.

## 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 + 4 + 8 + ⋯ and Grandi's series are divergent series and geometric series.

## Infinity

Infinity is something which is boundless, endless, or larger than any natural number.

## 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.

## 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.

## Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

## 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.

## Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z).

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. 1 + 2 + 4 + 8 + ⋯ and p-adic number are p-adic numbers.

## Power of two

A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. 1 + 2 + 4 + 8 + ⋯ and power of two are binary arithmetic.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n.

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

## Ramanujan summation

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

## Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

## Repeating decimal

A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity.

## 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.

## Two's complement

Two's complement is the most common method of representing signed (positive, negative, and zero) integers on computers, and more generally, fixed point binary values. 1 + 2 + 4 + 8 + ⋯ and Two's complement are binary arithmetic.

## 0.999...

In mathematics, 0.999... (also written as 0., 0., or 0.(9)) denotes the smallest number greater than every number in the sequence.

## 1 + 1 + 1 + 1 + ⋯

In mathematics,, also written,, or simply, is a divergent series. 1 + 2 + 4 + 8 + ⋯ and 1 + 1 + 1 + 1 + ⋯ are divergent series and geometric series.

## 1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series. 1 + 2 + 4 + 8 + ⋯ and 1 + 2 + 3 + 4 + ⋯ are divergent series.

## 1 − 1 + 2 − 6 + 24 − 120 + ⋯

In mathematics, is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. 1 + 2 + 4 + 8 + ⋯ and 1 − 1 + 2 − 6 + 24 − 120 + ⋯ are divergent series.

## 1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. 1 + 2 + 4 + 8 + ⋯ and 1 − 2 + 3 − 4 + ⋯ are divergent series.

## 1 − 2 + 4 − 8 + ⋯

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