A Novel Sixth-Root–Factorial Series Approximation for π Derived from (31)^(1/6)

Abstract

In this paper, we present a new analytical approximation for the mathematical constant π, derived from an unconventional relationship involving the sixth root of 31 and a rapidly convergent factorial-based infinite series. Starting from the equation

⁶(31)^(1/6) = √π + ∑(k = 1 to ∞) [ k / (8k)! ] − 1/9801,

we algebraically isolate √π and obtain an explicit symbolic expression for π. The proposed formulation combines radical expressions with a highly convergent factorial series, ensuring fast numerical stability and computational efficiency. A step-by-step derivation is provided, followed by numerical evaluation demonstrating that the resulting approximation yields π ≈ 3.14187, which is remarkably close to the accepted value of π ≈ 3.14159. The small deviation arises primarily from truncation of the infinite series, indicating that higher-order terms can further enhance accuracy. This work highlights a novel pathway for π-approximation, distinct from classical geometric, trigonometric, and Ramanujan-type series, and contributes to ongoing research in number theory and mathematical constants. The proposed method enriches the landscape of π-approximations and opens new directions for exploring factorial-series structures linked with radical expressions.