Skip to main content
SpringerLink
Log in

Concerning an infinite series of Ramanujan related to the natural logarithm

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)

    MATH  Google Scholar 

  2. Ball, K.: Completely monotonic rational functions and Hall’s marriage theorem. J. Combin. Theory Ser. B 61(1), 118–124 (1994). https://doi.org/10.1006/jctb.1994.1037

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndt, B.C.: Ramanujan’s Notebooks, Part IV. Springer, New York (1994)

    Book  Google Scholar 

  4. Borwein, J.M., Borwein, P.B.: The arithmetic–geometric mean and fast computation of elementary functions. SIAM Rev. 126(3), 351–366 (1984)

    Article  MathSciNet  Google Scholar 

  5. Borwein, J.M., Bradley, D.M., Crandall, R.E.: Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121(1–2), 247–296 (2000). https://doi.org/10.1016/S0377-0427(00)00336-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Bromwich, T .J .I’.A.: An Introduction to the Theory of Infinite Series, 3rd edn. Chelsea, New York (1991)

    MATH  Google Scholar 

  7. Day, W.A.: On monotonicity of the relaxation functions of vixcoelastic material. Proc. Camb. Philos. Soc. 67(2), 503–508 (1970). https://doi.org/10.1017/S0305004100045771P

    Article  Google Scholar 

  8. Frenzen, C.L.: Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18(3), 890–896 (1987). https://doi.org/10.1137/0518067

    Article  MathSciNet  MATH  Google Scholar 

  9. Gould, H.W.: Explicit formulas for Bernoulli numbers. Am. Math. Mon. 79(1), 44–51 (1972). https://doi.org/10.2307/2978125

    Article  MathSciNet  MATH  Google Scholar 

  10. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, London (2007)

    MATH  Google Scholar 

  11. Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M. (eds.): Collected Papers of Srinivasa Ramanujan. AMS Chelsea Publishing, Providence (2000)

  12. Kimberling, C.H.: A probabilistic interpretation of complete monotonicity. Aequ. Math. 10(2), 152–164 (1974). https://doi.org/10.1007/BF01832852

    Article  MathSciNet  MATH  Google Scholar 

  13. Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50, 527–628 (2013). https://doi.org/10.1090/S0273-0979-2013-01423-X

    Article  MathSciNet  MATH  Google Scholar 

  14. Lorch, L., Muldoon, M.E., Szego, P.: Higher monotonicity properties of certain Sturm–Liouville functions. III. Can. J. Math. 22, 1238–1265 (1970)

    Article  MathSciNet  Google Scholar 

  15. Mahajan, A., Ross, D.K.: A note on completely and absolutely monotone functions. Can. Math. Bull. 25(2), 143–148 (1982)

    Article  MathSciNet  Google Scholar 

  16. Olver, F.W.J., et al. (eds.): NIST Digital Library of Mathematical Functions. Release 1.013 of 16 Sept 2016. http://dlmf.nist.gov/24.2

  17. Ramanujan, S.: Notebooks (2 Volumes). Tata Institute of Fundamental Research, Bombay (1957)

    MATH  Google Scholar 

  18. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein functions: theory and applications. De Gruyter Studies in Mathematics, vol. 37, 2nd edn. Walter de Gruyter, Berlin (2012)

    Google Scholar 

  19. Sondow, J.: New Vacca-type rational series for Euler’s constant \(\gamma \) and its “alternating” analog \(\log 4/\pi \). In: Chudnovsky, D., Chudnovsky, G. (eds.) Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson, pp. 331–340. Springer, New York (2010). https://doi.org/10.1007/978-0-387-68361-4-23

    Chapter  Google Scholar 

  20. Sondow, J., Zudilin, W.: Euler’s constant, \(q\)-logarithms, and formulas of Ramanujan and Gosper. Ramanujan J. 12(2), 225–244 (2006). https://doi.org/10.1007/s11139-006-0075-1

    Article  MathSciNet  MATH  Google Scholar 

  21. Vacca, G.: A new series for the Eulerian constant \(\gamma =.577\dots \). Quart. J. Math. XL, 363–364 (1910)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, ME, 04469-5752, USA

    David M. Bradley

Authors
  1. David M. Bradley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David M. Bradley.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bradley, D.M. Concerning an infinite series of Ramanujan related to the natural logarithm. Ramanujan J 47, 253–265 (2018). https://doi.org/10.1007/s11139-017-9961-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-017-9961-y

Keywords

Mathematics Subject Classification

Search

Navigation