# 1/2 − 1/4 + 1/8 − 1/16 + ⋯

< 1 Demonstration that 1/21/4 + 1/81/16 + ⋯ = 1/3

In mathematics, the infinite series 1/21/4 + 1/81/16 + ⋯ is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2^n}=\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{\frac12}{1-(-\frac12)} = \frac13.$

## Hackenbush and the surreals

A slight rearrangement of the series reads

$1-\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac13.$

The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number 1/3:

LRRLRLR… = 1/3.

A slightly simpler Hackenbush string eliminates the repeated R:

LRLRLRL… = 2/3.

In terms of the Hackenbush game structure, this equation means that the board depicted on the right has a value of 0; whichever player moves second has a winning strategy.