Pi, Euler Numbers, and Asymptotic Expansions

  • J. M. Borwein
  • P. B. Borwein
  • K. Dilcher

Abstract

Gregory’s series for π, truncated at 500,000 terms, gives to forty places
$$4\sum\limits_{k = 1}^{500.000} {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{2k - 1}}} = 3.141590653589793240462643383269502884197$$

Keywords

Asymptotic Expansion Remainder Term Euler Number Summation Formula Analytic Number Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, N.Y., 1964.Google Scholar
  2. 2.
    M. D. Atkinson, How to compute the series expansions of sec x and tan x, Amer. Math. Monthly, 93 (1986) 387. 388.Google Scholar
  3. 3.
    B. C. Berndt, Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications, J. Number Theory, 7 ( 1975 413. 445.Google Scholar
  4. 4.
    J. M. Borwein and P. B. Borwein, Pi and the AGM - A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.Google Scholar
  5. 5.
    T. J. I’a Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed., MacMillan, London, 1926.Google Scholar
  6. 6.
    R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison Wesley, Reading, Mass., 1989.MATHGoogle Scholar
  7. 7.
    R. Johnsonbaugh, Summing an alternating series, this MONTHLY, 86 (1979) 637–648.MathSciNetMATHGoogle Scholar
  8. 8.
    D. E. Knuth and T. J. Buckholtz, Computation of Tangent, Euler, and Bernoulli numbers, Math. Comput.,21 (1967)663–688.Google Scholar
  9. 9.
    N. Nórlund, Vorlesungen fiber Differenzenrechnung, Springer-Verlag, Berlin, 1924.CrossRefGoogle Scholar
  10. 10.
    R. D. North, personal communications, 1988.Google Scholar
  11. 11.
    N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • J. M. Borwein
    • 1
  • P. B. Borwein
    • 1
  • K. Dilcher
    • 1
  1. 1.Dalhousie UniversityHalifaxCanada

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