1/2 − 1/4 + 1/8 − 1/16 + · · ·


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

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. [Berkelamp et al p.79]

A slightly simpler Hackenbush string eliminates the repeated R::LRLRLRL… = 2/3. [Berkelamp et al pp.307-308]

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. [Shawyer and Watson p.3]
*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. [See Korevaar p.325]

Notes

References


*cite book |author=Berlekamp, E.R.; J.H. Conway; and R.K. Guy |year=1982 |title=Winning Ways for your Mathematical Plays |publisher=Academic Press |id=ISBN 0-12-091101-9
*cite book |last=Korevaar |first=Jacob |title=Tauberian Theory: A Century of Developments |publisher=Springer |year=2004 |id=ISBN 3-540-21058-X
*cite book |author=Shawyer, Bruce and Bruce Watson |title=Borel's Methods of Summability: Theory and Applications |publisher=Oxford UP |year=1994 |id=ISBN 0-19-853585-6

ee also

1/2 + 1/4 + 1/8 + 1/16 + · · ·


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