Ramanujan-Type π Acceleration and the N-Series Geometric Constant πg

Ramanujan-Type π Acceleration and the N-Series Geometric Constant πg — v1.1

Author: C. Wayne Baker
Version: 1.1
Date: November 2, 2026
Research type: Expository mathematical preprint

Abstract

This paper presents a structural comparison between Ramanujan-type acceleration formulas for π and the N-Series geometric approximation framework. While Ramanujan's classical formulas achieve exceptionally rapid convergence through modular equations, elliptic integrals, and hypergeometric identities, the N-Series framework accelerates constructible polygonal approximations through systematic cancellation of successive even-power error terms.

Beginning with the regular inscribed polygon approximation

[
P_N=N\sin\left(\frac{\pi}{N}\right),
]

the paper develops two-scale and three-scale extrapolation operators that eliminate the leading asymptotic error terms. These yield convergence orders of (O(N^{-2})), (O(N^{-4})), and (O(N^{-6})) for the raw, two-scale, and three-scale approximations, respectively. Independent high-precision numerical calculations reproduce the predicted convergence orders.

Version 1.1 adds a quantitative comparison with Ramanujan's classical series for (1/\pi). The ratio of successive summands approaches (1/99^4), corresponding to approximately 7.98 decimal digits of asymptotic improvement per additional term. This convergence mechanism is distinguished explicitly from geometric N-Series extrapolation, whose convergence orders describe increasing polygonal resolution rather than digits gained per series term.

The geometrically defined constant (\pi_g) is shown to equal the classical constant (\pi). The construction does not introduce a new numerical constant, a new modular identity, or a method that computationally outperforms Ramanujan's series.

The contribution is an expository synthesis of two distinct mathematical acceleration mechanisms: modular-hypergeometric convergence and geometric scale cancellation.

Reproducibility

The accompanying archive contains the revised manuscript, LaTeX source, Python verification script, numerical results, reproducibility documentation, citation metadata, and SHA-256 checksums.

The numerical experiments use arbitrary-precision arithmetic to verify the predicted convergence orders.

Source repository: https://github.com/CWBaker360/ramanujan-pi-nseries-geometric-constant

Research status: Independently reproduced numerical results; mathematical arguments presented for scholarly review. No claim of independent external peer review is made.