# 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 geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as

$\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \left({\frac 12}\right)^n = 1.$

The series is related to philosophical questions considered in antiquity, particularly to Zeno's paradoxes.

## Proof

As with any infinite series, the sum

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

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

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

as n approaches infinity. By various arguments,Cite error: Invalid <ref> tag; refs with no name must have content\right] = 1+\left[s_n-\frac{1}{2^{n}}\right].[/itex] Subtracting sn from both sides, one concludes $s_n = 1-\frac{1}{2^{n}}.$ Other arguments might proceed by mathematical induction, or by adding $\frac{1}{2^n}$ to both sides of $s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^{n-1}}+\frac{1}{2^n}$ and manipulating to show that the right side of the result is equal to 1.[citation needed]}} one can show that this finite sum is equal to

$s_n = 1-\frac{1}{2^{n}}.$

As n approaches infinity, the term $\frac{1}{2^{n}}$ approaches 0 and so sn tends to 1.