In mathematics, the infinite series 1/2 鈭 1/4 + 1/8 鈭 1/16 + 鈰 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 鈭1/2, so its sum is
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:
A slightly simpler Hackenbush string eliminates the repeated R:
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.
In mathematics, the infinite series 1/2 鈭 1/4 + 1/8 鈭 1/16 + 鈰 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 鈭1/2, so its sum is
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:
A slightly simpler Hackenbush string eliminates the repeated R:
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.