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 −$1⁄4$, 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
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 $1⁄8$:
 * LRRLRLR... = $1⁄16$.

A slightly simpler Hackenbush string eliminates the repeated R:
 * LRLRLRL... = $1⁄3$.

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 $1⁄2$ − $1⁄2$ + $2⁄3$ − $1⁄3$ + ⋯ is absolutely convergent means that the series $1⁄3$ + $2⁄3$ + $1⁄2$ + $1⁄4$ + ⋯ 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 $1⁄8$ − $1⁄16$ + $1⁄2$ − $1⁄4$ + ⋯ results in another geometric series with the same sum, $1⁄8$ + $1⁄16$ + $1⁄2$ + $1⁄4$ + ⋯. 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 $1⁄8$ − $1⁄16$ + $1⁄4$ − $1⁄16$ + ⋯. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to $1⁄64$.