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

Direct proof

As with any infinite series, the infinite sum

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

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

Subtracting sn from both sides,

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]