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[[File:Eye of Horus square.png|thumb|First six summands drawn as portions of a square.]]
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In [[mathematics]], the [[infinite series]] '''1/2 + 1/4 + 1/8 + 1/16 + · · ·''' is an elementary example of a [[geometric series]] that [[absolute convergence|converges absolutely]].
 
Its [[geometric series#Sum|sum]] is
:<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \left({\frac 12}\right)^n = \frac {\frac12}{1-\frac 12} = 1. </math>
 
== Direct proof ==
 
As with any [[Series (mathematics)|infinite series]], the infinite sum
:<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots</math>
is defined to mean the [[Limit of a sequence|limit]] of the sum of the first {{mvar|n}} terms
:<math>s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^n}</math>
as {{mvar|n}} approaches infinity. Multiplying {{mvar|s<sub>n</sub>}} by 2 reveals a useful relationship:
:<math>2s_n = \frac22+\frac24+\frac28+\frac{2}{16}+\cdots+\frac{2}{2^n} = 1+\frac12+\frac14+\frac18+\cdots+\frac{1}{2^{n-1}} = 1+s_n-\frac{1}{2^n}.</math>
Subtracting {{mvar|s<sub>n</sub>}} from both sides,
:<math>s_n = 1-\frac{1}{2^n}.</math>
As {{mvar|n}} approaches infinity, {{mvar|s<sub>n</sub>}} [[Limit of a sequence|tends to]] 1.
 
==History==
This series was used as a representation of one of [[Zeno's paradoxes]].<ref>[http://web01.shu.edu/projects/reals/numser/series.html#zenonpdx Description of Zeno's paradoxes]</ref> The parts of the [[Eye of Horus#In arithmetic|Eye of Horus]] were once thought to represent the first six summands of the series.<ref>{{cite book | title=Professor Stewart's Hoard of Mathematical Treasures | last=Stewart | first=Ian | publisher=Profile Books | ISBN=978 1 84668 292 6 | year= 2009 | pages=76–80 }}</ref>
 
==See also==
*[[0.999...]]
*[[Dotted note]]
 
==References==
{{reflist}}
 
{{Series (mathematics)}}
 
{{DEFAULTSORT:1 2 + 1 4 + 1 8 + 1 16 +}}
[[Category:Geometric series]]
[[Category:One]]
 
 
{{mathanalysis-stub}}
 
[[zh:1/2+1/4+1/8+1/16+…]]

Revision as of 21:20, 9 February 2014

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