Revisiting the Formula for the Ramanujan Constant of a Series

Jocemar Q. Chagas, José A. Tenreiro Machado and António M. Lopes
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Jocemar Q. Chagas: Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil
José A. Tenreiro Machado: Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, 4249-015 Porto, Portugal
António M. Lopes: LAETA/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal

Mathematics, 2022, vol. 10, issue 9, 1-15

Abstract: 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.

Keywords: Ramanujan summation; Ramanujan constant of a series; divergent series; Euler–Maclaurin summation formula; Euler–Boole summation formula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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