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

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: A slightly simpler Hackenbush string eliminates the repeated R: 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.

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