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

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

Hackenbush and the surreals[]

Demonstration of 2/3 via a zero-value game

A slight rearrangement of the series reads

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

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.^[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.

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.^[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.^[4]

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