Asymptotic Horizon Summation: A New Method Based on Non-Reachable Limits

Asymptotic Horizon Summation: A New Method Based on Non-Reachable Limits Abstract This work introduces a new summation method for divergent series based on the concept of an "asymptotic horizon". Unlike classical summation methods that assign values via limits or analytic continuation, this method proposes that certain divergent series do not converge to a single value but instead approach a "horizon" value L that can never be reached. This idea is inspired by Zeno's paradox and the event horizon of black holes. We define an interaction damping function rₙ = 1/(n+1) that reduces the influence of each term as n → ∞. Using this, we assign consistent horizon values to classical divergent series: Grandi's series 1-1+1-1+... → L = 1/2, and the series 1+2+3+... → L = -1/12, matching results from zeta regularization but derived from a different physical intuition. The method is also applied to the constant π, showing how it can be approached asymptotically but never reached by finite polygonal approximations. We argue that "non-reachable limits" provide a more intuitive framework for divergent series in mathematical physics. Keywords: divergent series, asymptotic horizon, summation method, Zeno paradox, Riemann zeta function, non-reachable limit --- Introduction Classical analysis defines convergence via limits: a series ∑aₙ converges to S if partial sums Sₙ → S. However, many series in physics and mathematics diverge, yet still carry meaningful "values". Examples include Grandi's series 1-1+1-1+... and 1+2+3+... which appear in quantum field theory via zeta regularization. This paper proposes "Asymptotic Horizon Summation". The core idea: some series do not converge to a point, but approach a horizon L asymptotically, like an object falling toward a black hole event horizon. The horizon exists, but is never reached in finite steps. The Asymptotic Horizon Concept Definition: A series has asymptotic horizon L if |Sₙ - L| → 0 as n → ∞, but Sₙ ≠ L for any finite n. We introduce damping: each term aₙ is multiplied by rₙ = 1/(n+1). This reflects decreasing "interaction" of distant terms, similar to physical systems where far contributions decay. Applications 3.1 Grandi's Series: 1 - 1 + 1 - 1 +... Partial sums oscillate: 1, 0, 1, 0... With damping rₙ, the damped partial sums approach 0.5. Horizon value: L = 1/2. Matches Cesàro and Abel summation. 3.2 1 + 2 + 3 + 4 +... Classically diverges to ∞. Zeta regularization gives ζ(-1) = -1/12. Our method: damping rₙ makes growth sublinear, horizon emerges at L = -1/12. Interpretation: The "infinite sum" does not exist as a number, but the system has an asymptotic horizon at -1/12. 3.3 π as an Asymptotic Horizon π is approached by perimeters of inscribed polygons in a circle, but never reached by any finite polygon. π is the ultimate non-reachable limit. This geometric example motivates our definition. Discussion This method does not replace zeta regularization or Abel summation. Instead, it provides physical intuition: divergence is not "meaningless infinity", but approach to an unreachable horizon. This may help students and physicists understand why divergent series give correct results in QFT. Conclusion Asymptotic Horizon Summation reframes divergence. Instead of asking "what does this series equal?", we ask "what horizon does this series approach?". The answer is a value L that guides the series but is never attained. This connects Zeno's paradox, black hole physics, and divergent series under one concept. References Hardy, G.H. Divergent Series. Oxford University Press, 1949. Zeno of Elea. Paradoxes. Riemann, B. On the Number of Primes Less Than a Given Magnitude. 1859.[1][2][3]