Combinatorial Geometric Series and Negative Binomial Theorem: A Methodological Advance

Modern cryptographic algorithms and machine learning models often require efficient methods for handling discrete probability distributions and large-scale combinatorial data. This paper explores the Combinatorial Geometric Series (CGS) and its relationship to the Negative Binomial Theorem as a methodological advance for researchers in cybersecurity. This paper details how binomial expansions and Annamalai's identities can be used to optimize computing algorithms, particularly in the analysis of frequency distributions and error-correction codes.