1/2 + 1/4 + 1/8 + 1/16 + ⋯

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

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

Simple Proof

Let

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

Then

$2X = 2\frac12+2\frac14+2\frac18+2\frac{1}{16}+\cdots = 1+\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = 1+X.$

Thus

$2X = 1+X$ and

$X = 1.$

History

This series was used as a representation of one of Zeno's paradoxes. The parts of the Eye of Horus represent the first six summands of the series.

See also

• Dotted note