1 + 1 + 1 + 1 + ⋯

1/2-1/4+1/8

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

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

Hackenbush and the surreals

Demonstration of 2/3 via a zero-value game

A slight rearrangement of the series reads

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

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.

A slightly simpler Hackenbush string eliminates the repeated R:

LRLRLRL… = 2/3.

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.
• 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.

Notes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534