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Revisiting the Formula for the Ramanujan Constant of a Series

Author

Listed:
  • 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
    José A. Tenreiro Machado passed away.)

  • António M. Lopes

    (LAETA/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal)

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.

Suggested Citation

  • Jocemar Q. Chagas & José A. Tenreiro Machado & António M. Lopes, 2022. "Revisiting the Formula for the Ramanujan Constant of a Series," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1539-:d:808132
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    References listed on IDEAS

    as
    1. Jocemar Q. Chagas & José A. Tenreiro Machado & António M. Lopes, 2021. "Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums," Mathematics, MDPI, vol. 9(22), pages 1-38, November.
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    Cited by:

    1. Chunli Li & Wenchang Chu, 2022. "Evaluation of Infinite Series by Integrals," Mathematics, MDPI, vol. 10(14), pages 1-14, July.
    2. Mohammad Abu-Ghuwaleh & Rania Saadeh & Ahmad Qazza, 2022. "General Master Theorems of Integrals with Applications," Mathematics, MDPI, vol. 10(19), pages 1-19, September.

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