A Generating Function Approach to Finite Sum Identities in the Annamalai Combinatorial System

This paper provides a direct algebraic proof for the finite sum identity of combinatorial geometric series. Unlike inductive methods, this approach utilizes the decomposition of power series and the properties of generating functions to derive the closed-form expression. The result establishes a precise relationship between the truncated series and its infinite counterpart, offering significant computational advantages for real-time analytics and high-dimensional stochastic modeling.