A Methodological Framework for the Negative Binomial Theorem via Combinatorial Geometric Series and Generating Functions

This paper introduces a novel methodological advancement in the derivation of the Negative Binomial Theorem through the application of Combinatorial Geometric Series (CGS). By leveraging iterative summations of basic geometric series, the proposed framework establishes a direct mathematical bridge between combinatorial enumeration and generating functions. The research explores the convergence properties of the series, distinguishing between Power Series and Reciprocal Laurent Series based on variable magnitude to ensure numerical stability in diverse computational environments. This approach provides a closed-form expression that maps efficiently onto algorithmic structures, offering significant improvements for stochastic modeling in cryptography, machine learning, and high-dimensional data analytics.