1/2 − 1/4 + 1/8 − 1/16 + ⋯ Explained
In mathematics, the infinite series is a simple example of an alternating series that converges absolutely.
It is a geometric series whose first term is and whose common ratio is −, so its sum is
Hackenbush and the surreals
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 :
LRRLRLR... = .
A slightly simpler Hackenbush string eliminates the repeated R:
LRLRLRL... = .
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 is absolutely convergent means that the series 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 results in another geometric series with the same sum, . 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 is . Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to .
References
- Book: Berlekamp . E. R. . Elwyn Berlekamp . Conway . J. H. . John Horton Conway . Guy . R. K. . Richard K. Guy . . 1982 . Academic Press . 0-12-091101-9.
- Book: Korevaar, Jacob
. Tauberian Theory: A Century of Developments . Springer . 2004 . 3-540-21058-X.
- Book: Shawyer . Bruce . Watson . Bruce . Borel's Methods of Summability: Theory and Applications . Oxford UP . 1994 . 0-19-853585-6.
