Divergent Infinite Series -- Ramanujan's Initial Intuition

Abstract

This paper investigates Srinivasa Ramanujan's initial intuitive methodology for assigning the finite value -1/12 to the sum of the divergent infinite series of all positive integers. We systematically examine Ramanujan's initial method, originally sketched in his notebooks, and set the methodology into an algebraic framework. The methodology has limited applicability to other classes of divergent series. The methodology is extended to assign a Ramanujan smoothed sum to the infinite sequences of integers raised to a positive integer power and to figurate binomial number sequences, including triangular numbers, tetrahedral numbers, and higher-dimensional analogues, avoiding analytical continuation. A key finding establishes that the Ramanujan smoothed sums of figurate binomial sequences are intrinsically connected to logarithmic numbers (Gregory coefficients), providing a novel perspective on Ramanujan summation through the lens of classical combinatorial functions. The paper applies asymptotic expansions of associated rational generating functions to demonstrate consistency with established results from analytic continuation methods. The results illuminate the deeper mathematical structures underlying Ramanujan's intuitive insights and suggest new avenues for research in divergent series summation.