1/2 + 1/4 + 1/8 + 1/16 + ⋯

First six summands drawn as portions of a square.

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.

Its sum is

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

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

${\displaystyle s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n-1}}}+{\frac {1}{2^{n}}}}$

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

${\displaystyle 2s_{n}={\frac {2}{2}}+{\frac {2}{4}}+{\frac {2}{8}}+{\frac {2}{16}}+\cdots +{\frac {2}{2^{n}}}=1+\left[{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n-1}}}\right]=1+\left[s_{n}-{\frac {1}{2^{n}}}\right].}$

Subtracting s_n from both sides,

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

As n approaches infinity, s_n 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

• 0.999...
• Dotted note
• Eye of Horus#Mathematics

References

1. Wachsmuth, Bet G. "Description of Zeno's paradoxes". Archived from the original on 2015-12-31. Retrieved 2014-12-29.
2. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN 978 1 84668 292 6.