1/2 + 1/4 + 1/8 + 1/16 + ⋯

Infinite series summable to 1 thumb|First six summands drawn as portions of a square. thumb|The geometric series on the real line. In mathematics, the infinite series 1 2 + 1 4 + 1 8 + 1 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as The series is related to philosophical questions considered in antiquity, particularly to Zeno's paradoxes. Proof As with any infinite series, the sum is defined to mean the limit of the partial sum of the first n terms as n approaches infinity, if it exists. By various arguments,For example: multiplying sn by 2 yields Subtracting sn from both sides, one concludes Other arguments might proceed by mathematical induction, or by adding to both sides of and manipulating to show that the right side of the result is equal to 1.November 2020Abramowitz, M.; Stegun, I. A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9th printing). New York: Dover. p. 10. one can show that each finite sum is equal to and as n approaches infinity, the term approaches 0 and so sn approaches 1. History Zeno's paradox This series was used as a representation of two of Zeno's paradoxes.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. Similarly, Zeno's dichotomy paradox arises from the supposition that to move a certain distance, one would 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 In both cases, each time interval is a term of this infinite geometric series, and so even in the limit of infinite terms it would sum to a finite total time. This is sometimes considered to resolve Zeno's paradoxes.2010 Zeno's Paradoxes: 5. Zeno's Influence on Philosophy Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/paradox-zeno/#ZenInf 2011-03-07 Huggett Nick https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#ZenInf 2022-03-01 live However, insofar as Zeno was concerned with the problems of division of a continuum into an actual infinity of sub-parts, rather than the problem of their sum, it may not address the philosophical heart of Zeno's argument.Papa-Grimaldi Alba 1996 Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf live The Review of Metaphysics 50 299–314 https://web.archive.org/web/20120609113959/http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf 2012-06-09 2012-03-06 The Eye of Horus The parts of the Eye of Horus were once thought to represent the first six summands of the series.Professor Stewart's Hoard of Mathematical Treasures Stewart Ian Profile Books 978-1-84668-292-6 2009 76–80 In Zhuangzi A version of the series appears in the ancient Taoist book Zhuangzi. The miscellaneous chapters "All Under Heaven" include the following sentence: "Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted."Fraser Chris School of Names > Miscellaneous Paradoxes (Stanford Encyclopedia of Philosophy) https://plato.stanford.edu/entries/school-names/paradoxes.html plato.stanford.edu 24 July 2025 en See also 0.999... 1/2 − 1/4 + 1/8 − 1/16 + ⋯ Actual infinity Notes References Category:Geometric series Category:1 (number)