Separation principle: Difference between revisions
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[[File:Eye of Horus square.png|thumb|First six summands drawn as portions of a square.]] | |||
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 12:47, 1 October 2013
In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a geometric series that converges absolutely.
Its sum is
Direct proof
As with any infinite series, the infinite sum
is defined to mean the limit of the sum of the first Template:Mvar terms
as Template:Mvar approaches infinity. Multiplying Template:Mvar by 2 reveals a useful relationship:
Subtracting Template:Mvar from both sides,
As Template:Mvar approaches infinity, Template:Mvar tends to 1.
History
This series was used as a representation of one of Zeno's paradoxes.[1] The parts of the Eye of Horus were once thought to represent the first six summands of the series.[2]
See also
References
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- ↑ Description of Zeno's paradoxes
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