1/2 + 1/4 + 1/8 + 1/16 + ⋯

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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 converges absolutely.

Its sum is

$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=0}^\infty \frac12\left({\frac 12}\right)^n = \frac {\frac12}{1-\frac 12} = 1.$

Direct proof

As with any infinite series, the infinite sum

$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots$

is defined to mean the limit of the sum of the first n terms

$s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^n}$

as n approaches infinity. Multiplying sn by 2 reveals a useful relationship:

$2s_n = \frac22+\frac24+\frac28+\frac{2}{16}+\cdots+\frac{2}{2^n} = 1+\frac12+\frac14+\frac18+\cdots+\frac{1}{2^{n-1}} = 1+s_n-\frac{1}{2^n}.$

Subtracting sn from both sides,

$s_n = 1-\frac{1}{2^n}.$

As n approaches infinity, sn 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]

References

1. ^ Description of Zeno's paradoxes
2. ^ Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN 978 1 84668 292 6.