Understanding the Infinite Series: 1/2 − 1/4 + 1/8 − 1/16 + ⋯
In mathematics, the infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ serves as a fundamental example of an alternating series that converges absolutely. This series is classified as a geometric series with its first term being 1/2 and a common ratio of −1/2. Consequently, the sum can be derived through the formula for the sum of a geometric series.
The mathematical representation of the series is as follows:
∑n=1∞ (−1)n+1 / 2n = 1/2 − 1/4 + 1/8 − 1/16 + ⋯ = 1/3
This shows that the converged sum of the series is 1/3.
Exploring Hackenbush and Surreal Numbers
A slight rearrangement of the series can be expressed as:
1 − 1/2 − 1/4 + 1/8 − 1/16 + ⋯ = 1/3
The series can be interpreted through the lens of the Hackenbush game, where it is represented as an infinite blue-red Hackenbush string corresponding to the surreal number 1/3:
LRRLRLR… = 1/3
A simplified version of the Hackenbush string, which eliminates repeated moves, is:
LRLRLRL… = 2/3
This representation indicates that within the game structure, the board depicted has a value of 0, thus granting the second player a winning strategy.
Related Series and Historical Context
The implication that 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is absolutely convergent indicates that the series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is also convergent. This latter series converges to 1 and provides insight into one of the binary expansions of 1, expressed as 0.111….
When pairing terms of the series 1/2 − 1/4 + 1/8 − 1/16 + ⋯, we acquire another geometric series with an identical sum, 1/4 + 1/16 + 1/64 + 1/256 + ⋯. This series is noteworthy as one of the first to be summed throughout the history of mathematics, utilized by Archimedes circa 250–200 BC.
Furthermore, the Euler transform applied to the divergent series 1 − 2 + 4 − 8 + ⋯ results in the convergent series 1/2 − 1/4 + 1/8 − 1/16 + ⋯, illustrating that despite the former series lacking a traditional sum, it is Euler summable to 1/3.
Conclusion
The exploration of the infinite series and its mathematical nuances illustrates the profound connections between series, games, and numerical representations, enriching our understanding of mathematical convergence and surreal numbers.