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

{\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

A slight rearrangement of the series reads

1-{\frac 12}-{\frac 14}+{\frac 18}-{\frac {1}{16}}+\cdots ={\frac 13}.

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]

Notes

↑ Berkelamp et al. p. 79
↑ Berkelamp et al. pp. 307–308
↑ Shawyer and Watson p. 3
↑ Korevaar p. 325

References

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

Sequences and series

Integer
sequences

Basic	Arithmetic progression Geometric progression Square number Cubic number Factorial Powers of two Powers of 10

Advanced	Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number others

Properties of sequences

Properties of series

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/1^s + 1/2^s+ 1/3^s + ... (Riemann zeta function)

Divergent	1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric
series

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