# 1/2 − 1/4 + 1/8 − 1/16 + ⋯

In mathematics, the infinite series **1/2 − 1/4 + 1/8 − 1/16 + ⋯** 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

## Hackenbush and the surreals[edit]

A slight rearrangement of the series reads

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.

## Related series[edit]

- The statement that 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is absolutely convergent means that the series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111….
- Pairing up the terms of the series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ results in another geometric series with the same sum, 1/4 + 1/16 + 1/64 + 1/256 + ⋯. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.
^{[3]} - The Euler transform of the divergent series 1 − 2 + 4 − 8 + ⋯ is 1/2 − 1/4 + 1/8 − 1/16 + ⋯. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to 1/3.
^{[4]}

## Notes[edit]

## References[edit]

- 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.