1/2 − 1/4 + 1/8 − 1/16 + ⋯

Infinite series summable to 1/3 frame|right|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 Hackenbush and the surreals frame|right|Demonstration of 2 3 via a zero-value game A slight rearrangement of the series reads 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.Berlekamp Conway Guy 1982 79 A slightly simpler Hackenbush string eliminates the repeated R: LRLRLRL... = 2 3.Berlekamp Conway Guy 1982 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 Watson 1994 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.Korevaar 2004 325 Notes References Berlekamp E. R. Elwyn Berlekamp Conway J. H. John Horton Conway Guy R. K. Richard K. Guy Winning Ways for your Mathematical Plays 1982 Academic Press 0-12-091101-9 Korevaar Jacob Tauberian Theory: A Century of Developments https://archive.org/details/springer_10.1007-978-3-662-10225-1 Springer 2004 3-540-21058-X Shawyer Bruce Watson Bruce Borel's Methods of Summability: Theory and Applications Oxford UP 1994 0-19-853585-6 Category:Geometric series