A Bouquet of Series

  • Ovidiu Furdui
Part of the Problem Books in Mathematics book series (PBM)


This chapter offers the reader a bouquet of problems with a flavor towards the computational aspects of infinite series and special products, many of these problems being new in the literature. These series, linear or quadratic, single or multiple, involve combinations of exotic terms, special functions, and harmonic numbers and challenge the reader to explore the ability to evaluate an infinite sum, to discover new connections between a series and an integral, to evaluate a sum by using the modern tools of analysis, and to investigate further.


Alternative Solution Linear Differential Equation Product Formula Harmonic Number Harmonic Series 


  1. 5.
    Andreoli, M.: Problem 819, Problems and Solutions. Coll. Math. J. 37(1), 60 (2006)MathSciNetGoogle Scholar
  2. 7.
    Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)MATHGoogle Scholar
  3. 8.
    Basu, A., Apostol, T.M.: A new method for investigating Euler sums. Ramanujan J. 4, 397–419 (2000)MathSciNetMATHCrossRefGoogle Scholar
  4. 9.
    Bartle, R.G.: The Elements of Real Analysis, 2nd edn. Wiley, New York (1976)Google Scholar
  5. 10.
    Bataille, M.: Problem 1784, Problems and Solutions. Math. Mag. 81(5), 379 (2008)Google Scholar
  6. 11.
    Bataille, M., Cibes, M., G.R.A.20 Problem Solving Group, Seaman, W.: Problem 844, Problems and Solutions. Coll. Math. J. 39(1), 71–72 (2008)Google Scholar
  7. 12.
    Benito, M., Ciaurri, O.: Problem 3301. Crux with Mayhem 35(1), 47–49 (2009)Google Scholar
  8. 14.
    Biler, P., Witkowski, A.: Problems in Mathematical Analysis. Marcel Dekker, New York (1990)MATHGoogle Scholar
  9. 16.
    Boros, G., Moll, V.H.: Irresistible Integrals, Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004)MATHCrossRefGoogle Scholar
  10. 17.
    Borwein D., Borwein J.: On an intriguing integral and some series related to ζ(4). Proc. Am. Math. Soc. 123(4), 1191–1198 (1995)MATHGoogle Scholar
  11. 18.
    Boyadzhiev, K.N.: Problem H-691, Advanced Problems and Solutions. Fibonacci Q. 50(1), 90–92 (2012)Google Scholar
  12. 19.
    Boyadzhiev, K.N.: On a series of Furdui and Qin and some related integrals, p. 7, August 2012. Available at http://arxiv.org/pdf/1203.4618.pdf
  13. 20.
    Bromwich, T.J.I’A.: An Introduction to the Theory of Infinite Series, 3rd edn. AMS Chelsea Publishing, Providence (1991)Google Scholar
  14. 23.
    Chen, H.: Problem 854, Problems and Solutions. Coll. Math. J. 39(3), 243–245 (2008)Google Scholar
  15. 24.
    Chen, H.: Evaluation of some variant Euler sums. J. Integer Seq. 9, Article 06.2.3 (2006)Google Scholar
  16. 25.
    Choi, J., Srivastava, H.M.: Explicit evaluation of Euler and related sums. Ramanujan J. 10, 51–70 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 26.
    Coffey, M.W.: Integral and series representation of the digamma and polygamma functions, p. 27, August 2010. Available at http://arxiv.org/pdf/1008.0040.pdf
  18. 27.
    Coffey, M.W.: Expressions for two generalized Furdui series. Analysis, Munchen 31(1), 61–66 (2011)MathSciNetMATHCrossRefGoogle Scholar
  19. 29.
    Comtet, L.: Advanced Combinatorics, Revised and Enlarged Edition. D. Reidel Publishing Company, Dordrecht (1974)MATHCrossRefGoogle Scholar
  20. 35.
    Dobiński, G.: Summirung der Reihe ∑n m ∕ n! für m = 1, 2, 3, 4, 5, . Arch. für Mat. und Physik 61, 333–336 (1877)Google Scholar
  21. 38.
    Dwight, H.B.: Table of Integrals and Other Mathematical Data. The Macmillan Company, New York (1934)Google Scholar
  22. 40.
    Efthimiou, C.: Remark on problem 854, Problems and Solutions. Coll. Math. J. 40(2), 140–141 (2009)Google Scholar
  23. 43.
    Finch, S.R.: Mathematical Constants. Encyclopedia of Mathematics and Its Applications 94. Cambridge University Press, New York (2003)Google Scholar
  24. 44.
    Freitas, P., Integrals of Polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74(251), 1425–1440 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. 46.
    Furdui, O.: A convergence criteria for multiple harmonic series. Creat. Math. Inform. 18(1), 22–25 (2009)MathSciNetMATHGoogle Scholar
  26. 47.
    Furdui, O.: Series involving products of two harmonic numbers. Math. Mag. 84(5), 371–377 (2011)MathSciNetMATHCrossRefGoogle Scholar
  27. 52.
    Furdui, O.: Closed form evaluation of a multiple harmonic series. Automat. Comput. Appl. Math. 20(1), 19–24 (2011)MathSciNetGoogle Scholar
  28. 54.
    Furdui, O., Trif, T.: On the summation of certain iterated series. J. Integer Seq. 14, Article 11.6.1 (2011)Google Scholar
  29. 55.
    Furdui, O., Qin, H.: Problema 149, Problemas y Soluciones. Gac. R. Soc. Mat. Esp. 14(1), 103–106 (2011)Google Scholar
  30. 62.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic, San Diego (2000)MATHGoogle Scholar
  31. 64.
    Harms, P.M.: Problem 5097, Solutions. Sch. Sci. Math. 110(4), 15–16 (2010)Google Scholar
  32. 66.
    Herman, E.: Problem 920, Problems and Solutions. Coll. Math. J. 42(1), 69–70 (2011)Google Scholar
  33. 70.
    Janous, W.: Around Apery’s constant. JIPAM, J. Inequal. Pure Appl. Math. 7(1), Article 35 (2006)Google Scholar
  34. 73.
    Klamkin, M.S.: A summation problem, Advanced Problem 4431. Am. Math. Mon. 58, 195 (1951); Am. Math. Mon. 59, 471–472 (1952)Google Scholar
  35. 74.
    Klamkin, M.S.: Another summation. Am. Math. Mon. 62, 129–130 (1955)CrossRefGoogle Scholar
  36. 75.
    Boyd, J.N., Hurd, C., Ross, K., Vowe, M.: Problem 475, Problems and Solutions. Coll. Math. J. 24(2), 189–190 (1993)Google Scholar
  37. 77.
    Kouba, O.: Problem 1849, Problems and Solutions. Math. Mag. 84(3), 234–235 (2011)Google Scholar
  38. 78.
    Kouba, O.: The sum of certain series related to harmonic numbers, p. 14, March 2012. Available at http://arxiv.org/pdf/1010.1842.pdf
  39. 79.
    Kouba, O.: A Harmonic Series, SIAM Problems and Solutions, Classical Analysis, Sequences and series, Problem 06-007 (2007). http://www.siam.org/journals/categories/06-007.php
  40. 80.
    Krasopoulos, P.T.: Problem 166, Solutions. Missouri J. Math. Sci. 20(3) (2008)Google Scholar
  41. 83.
    Lau, K.W.: Problem 5013, Solutions. Sch. Sci. Math. 108(6) (2008)Google Scholar
  42. 84.
    Lewin, L.: Polylogarithms and Associated Functions. Elsevier (North-Holand), New York (1981)MATHGoogle Scholar
  43. 86.
    Mabry, R.: Problem 893, Problems and Solutions. Coll. Math. J. 41(1), 67–69 (2010)MathSciNetGoogle Scholar
  44. 89.
    Miller, A., Refolio, F.P.: Problem 895, Problems and Solutions. Coll. Math. J. 41(1), 70–71 (2010)Google Scholar
  45. 90.
    Mortici, C.: Problem 3375, Solutions. Crux with Mayhem 35(6), 415–416 (2009)Google Scholar
  46. 97.
    Ogreid, M.O., Osland, P.: More series related to the Euler series. J. Comput. Appl. Math. 136, 389–403 (2001)MathSciNetMATHCrossRefGoogle Scholar
  47. 98.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: In: Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.) NIST Handbook of Mathematical Functions. NIST National Institute of Standards and Technology, US Department of Commerce and Cambridge University Press, Cambridge (2010)Google Scholar
  48. 103.
    Plaza, Á.: Problem 2011-2, Electronic Journal of Differential Equations, Problem section (2011). Available electronically at http://math.uc.edu/ode/odesols/s2011-2.pdf
  49. 104.
    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis, vol. 1. Verlag von Julius Springer, Berlin (1925)MATHGoogle Scholar
  50. 107.
    Rassias, M.Th.: On the representation of the number of integral points of an elliptic curve modulo a prime number, p. 17, October 2012. Available at http://arxiv.org/pdf/1210.1439.pdf
  51. 108.
    Rădulescu, T.L., Rădulescu, D.V., Andreescu, T.: Problems in Real Analysis: Advanced Calculus on the Real Axis. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  52. 109.
    Remmert, R.: Classical Topics in Complex Function Theory. Graduate Text in Mathematics, vol. 172. Springer, New York (1998)Google Scholar
  53. 115.
    Seaman, W., Vowe, M.: Problem 906, Problems and Solutions. Coll. Math. J. 41(4), 330–331 (2010)Google Scholar
  54. 116.
    Seiffert, H.-J.: Problem H-653, Advanced Problems and Solutions. Fibonacci Q. 46/47(2), 191–192 (2009)Google Scholar
  55. 120.
    Sondow, J.: Double integrals for Euler’s constant and ln4 ∕ π and an analogue of Hadjicostas’s formula. Am. Math. Mon. 112, 61–65 (2005)MathSciNetMATHCrossRefGoogle Scholar
  56. 122.
    Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer, Dordrecht (2001)MATHCrossRefGoogle Scholar
  57. 123.
    Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Insights, Amsterdam (2012)MATHGoogle Scholar
  58. 127.
    Stong, R.: Problem 11371, Problems and Solutions. Am. Math. Mon. 117(5), 461–462 (2010)Google Scholar
  59. 128.
    Summer, J., Kadic-Galeb, A.: Problem 873, Problems and Solutions. Coll. Math. J. 40(2), 136–137 (2009)Google Scholar
  60. 130.
    Tyler, D.B.: Problem 11494, Problems and Solutions. Am. Math. Mon. 118(9), 850–851 (2011)Google Scholar
  61. 131.
    Vara, G.C.J.: Problema 94, Problemas y Soluciones. Gac. R. Soc. Mat. Esp. 11(4), 698–701 (2008)Google Scholar
  62. 133.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1927)MATHGoogle Scholar
  63. 134.
    Young, R.M.: Euler’s constant. Math. Gaz. 472, 187–190 (1991)CrossRefGoogle Scholar
  64. 138.
    Zhu, K.: Analysis on Fock Spaces. Graduate Text in Mathematics, vol. 263. Springer, New York (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ovidiu Furdui
    • 1
  1. 1.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania

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