infinite series 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. The study of series is a major part of calculus and its generalization, math ...

is an elementary example of a

geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...

that

converges absolutely In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...

. The sum of the series is 1. In summation notation, this may be expressed as :

\frac12+\frac14+\frac18+\frac+\cdots = \sum_^\infty \left(\right)^n = 1.

The series is related to philosophical questions considered in antiquity, particularly to

Zeno's paradoxes Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plural ...

Proof

As with any

, the sum :

\frac12+\frac14+\frac18+\frac+\cdots

is defined to mean the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of the

partial sum 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. The study of series is a major part of calculus and its generalization, math ...

of the first terms :

s_n=\frac12+\frac14+\frac18+\frac+\cdots+\frac+\frac

as approaches infinity. By various arguments, one can show that this finite sum is equal to :

s_n = 1-\frac.

As approaches infinity, the term

\frac

approaches 0 and so tends to 1.

History

Zeno's paradox

This series was used as a representation of many of

.Field, Paul and Weisstein, Eric W. "Zeno's Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZenosParadoxes.html For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but the tortoise would already have moved another five meters by then. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on. Zeno argued that the tortoise would always remain ahead of Achilles. The

Dichotomy paradox Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in pluralit ...

also states that to move a certain distance, you have to move half of it, then half of the remaining distance, and so on, therefore having infinitely many time intervals.Field, Paul and Weisstein, Eric W. "Zeno's Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZenosParadoxes.html This can be easily resolved by noting that each time interval is a term of the

infinite geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...

, and will sum to a finite number.

The Eye of Horus

The parts of the Eye of Horus were once thought to represent the first six summands of the series.

In a myriad ages it will not be exhausted

A version of the series appears in the ancient Taoist book ''

Zhuangzi Zhuangzi may refer to: * ''Zhuangzi'' (book) (莊子), an ancient Chinese collection of anecdotes and fables, one of the foundational texts of Daoism **Zhuang Zhou Zhuang Zhou (), commonly known as Zhuangzi (; ; literally "Master Zhuang"; als ...

''. The miscellaneous chapters "All Under Heaven" include the following sentence: "Take a

chi Chi or CHI may refer to: Greek *Chi (letter), the Greek letter (uppercase Χ, lowercase χ); Chinese *Chi (length), ''Chi'' (length) (尺), a traditional unit of length, about ⅓ meter *Chi (mythology) (螭), a dragon *Chi (surname) (池, pin ...

long stick and remove half every day, in a myriad ages it will not be exhausted."

Notes

References

{{DEFAULTSORT:1 2 + 1 4 + 1 8 + 1 16 + Geometric series 1 (number)