1/2 − 1/4 + 1/8 − 1/16 + ⋯

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

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2^{n}}}={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {\frac {1}{2}}{1-(-{\frac {1}{2}})}}={\frac {1}{3}}.}$

Hackenbush and the surreals

Demonstration of 2/3 via a zero-value game

A slight rearrangement of the series reads

${\displaystyle 1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {1}{3}}.}$

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.[1]

A slightly simpler Hackenbush string eliminates the repeated R:

LRLRLRL… = 2/3.[2]

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.

Notes

1. ^ Berkelamp et al. p. 79
2. ^ Berkelamp et al. pp. 307–308
3. ^ Shawyer and Watson p. 3
4. ^ Korevaar p. 325