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

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

Hackenbush and the surrealsEdit

Demonstration of 2/3 via a zero-value game

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 seriesEdit


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


  • 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.