1/2 + 1/4 + 1/8 + 1/16 + · · ·

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%2B\frac14%2B\frac18%2B\frac{1}{16}%2B\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) %2B \frac{1}{2}\left( \frac{1}{2} \right) %2B \cdots =\sum_{n=0}^{\infty} \frac{1}{2} \left( \frac{1}{2} \right) ^n

This is a special case of the geometric series:

a %2B ar %2B ar^2 %2B \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

  1. ^ Description of Zeno's paradoxes

See also