1/2 + 1/4 + 1/8 + 1/16 + · · ·

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In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a series that converges absolutely.

Its sum is

$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \frac{1}{2^n}$

By factoring 1/2 out of every term in this series this infinite sum can be rewritten as:

$\frac{1}{2}\left( 1 \right) + \frac{1}{2}\left( \frac{1}{2} \right) + \cdots =\sum_{n=0}^{\infty} \frac{1}{2} \left( \frac{1}{2} \right) ^n$

This is a special case of the geometric series:

$a + ar + ar^2 + \cdots=\sum_{n=0}^{\infty}ar^n$

where a is the common ratio (gcd) between all terms

These series are known to converge to a finite point given the following condition:

$r \in \ ]-1,1[$

If a series can be written in the general form stated above and satisfies the preceding condition the series converges to a computable value at infinity given by the following formula:

$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$

where a is the common ratio

Considering our special case where:

$a=\frac{1}{2}, r=\frac{1}{2}$

We yield the following result:

$\sum_{n=0}^{\infty} \frac{1}{2} \left( \frac{1}{2} \right) ^n = \frac{\frac{1}{2}}{1-\frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$

History

This series was used as a representation of one of Zeno's paradoxes.^[1] The parts of the Eye of Horus represent the first six summands of the series.

Notes

^ Description of Zeno's paradoxes

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