1/2 − 1/4 + 1/8 − 1/16 + ⋯

Demonstration that Template:Sfrac − Template:Sfrac + Template:Sfrac − Template:Sfrac + ⋯ = Template:Sfrac

In mathematics, the infinite series Template:Nowrap is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is Template:Sfrac and whose common ratio is −Template:Sfrac, so its sum is

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

Hackenbush and the surreals

Demonstration of Template:Sfrac via a zero-value game

A slight rearrangement of the series reads

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

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 Template:Sfrac:

LRRLRLR... = Template:Sfrac.Template:Sfn

A slightly simpler Hackenbush string eliminates the repeated R:

LRLRLRL... = Template:Sfrac.Template:Sfn

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 Template:Nowrap is absolutely convergent means that the series Template:Nowrap 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 Template:Nowrap results in another geometric series with the same sum, Template:Nowrap. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.Template:Sfn
• The Euler transform of the divergent series Template:Nowrap is Template:Nowrap. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to Template:Sfrac.Template:Sfn

References

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