1/2 − 1/4 + 1/8 − 1/16 + ⋯

Demonstration that ⁠1/2⁠ − ⁠1/4⁠ + ⁠1/8⁠ − ⁠1/16⁠ + ⋯ = ⁠1/3⁠

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

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

Hackenbush and the surreals

Demonstration of ⁠2/3⁠ via a zero-value game

A slight rearrangement of the series reads

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

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.

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.

Related series

• 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.
• 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.

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.