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Sharp Bounds for the Constant e

Abstract

We show validity of \(\frac {1}{(n+\alpha )n!}\leq \mathrm {e}-\sum \nolimits _{k=0}^{n}\frac {1}{k!}<\frac {1}{(n+\beta )n!}\) for any integer n ≥ 1, with the best possible constants \(\alpha =\frac {1}{\mathrm {e}-2}-1\approxeq 0.39\) and β = 0; furthermore, we obtain a more precise form of this inequality.

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Acknowledgements

We express our gratitude to the referees for mentioning some valuable comments on the historical background of the paper.

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Correspondence to Mehdi Hassani.

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Hassani, M., Sofo, A. Sharp Bounds for the Constant e . Vietnam J. Math. 43, 629–633 (2015). https://doi.org/10.1007/s10013-014-0114-y

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  • DOI: https://doi.org/10.1007/s10013-014-0114-y

Keywords

  • The number e
  • Approximation
  • Best constants
  • Special functions

Mathematics Subject Classification (2010)

  • 40A05
  • 41A44
  • 41A30