1/2 + 1/4 + 1/8 + 1/16 + ⋯

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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[edit]

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.


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[edit]


  1. ^ Wachsmuth, Bet G. "Description of Zeno's paradoxes". Archived from the original on 2014-12-31. 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.