Revisiting the Formula for the Ramanujan Constant of a Series

The main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the Euler-Boole summation formula. Such an expression is not present in the literature. We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods for divergent series. Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler-Maclaurin summation formula. Finally, we present several examples of the Ramanujan constant of divergent series.

Automated summary

The main contribution of this paper is the proposal of a closed expression for the Ramanujan constant of alternating series based on the Euler-Boole summation formula. It also highlights the unique choice for the parameter a in Hardy's formula for a series of positive terms to obtain a Ramanujan constant that agrees with other summation methods for divergent series. The paper further derives a closed-formula for the Ramanujan constant of a series with the chosen parameter, under a natural interpretation of the integral term in the Euler-Maclaurin summation formula. Several examples of the Ramanujan constant of divergent series are presented.