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

$\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13.$

Hackenbush and the surreals

Demonstration of 2/3 via a zero-value game

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 1/3:

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.

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Related series

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Notes

1. ^ Berkelamp et al. p.79
2. ^ Berkelamp et al. pp.307-308
3. ^ Shawyer and Watson p.3
4. ^ See Korevaar p.325
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References

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