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

${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}={\frac {\frac {1}{2}}{1-{\frac {1}{2}}}}=1.$$\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

${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots$$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots$

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

$s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n}}}$$s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^n}$

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

$2s_{n}={\frac {2}{2}}+{\frac {2}{4}}+{\frac {2}{8}}+{\frac {2}{16}}+\cdots +{\frac {2}{2^{n}}}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n-1}}}=1+s_{n}-{\frac {1}{2^{n}}}.$$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 s_n from both sides,

$s_{n}=1-{\frac {1}{2^{n}}}.$$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. The parts of the Eye of Horus were once thought to represent the first six summands of the series.

See also

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