# 1/2 − 1/4 + 1/8 − 1/16 + ⋯

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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

Demonstration of 2/3 via a zero-value game

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.[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

## References

• Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 0-12-091101-9.
• Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.