Continued reciprocal roots

Abstract

For \(p\) a real number, a continued \(p\)th power is an expression of the form \(\lim _{n\rightarrow \infty } a_0+(a_1+(a_2+(\ldots +(a_n)^p\ldots )^p)^p)^p.\) Continued reciprocal roots are the special case in which \(-1<p<0\). Following a brief historical survey, we derive a sharp necessary and sufficient divergence condition for continued reciprocal \(q\)th roots of positive terms, where \(q=|1/p|\).

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

Fig. 1

Notes

  1. 1.

    Even in 1877, Réalis [6] could dismiss a long paper from 1862 [3] with the remark “[L]a théorie des radicaux continus est loin d’être nouvelle, et M. A. Bouché n’est pas le premier qui s’en soit occupé. [The theory of continued radicals is far from new, and Monsieur A. Bouché is not the first to have studied it.]”

  2. 2.

    Indeed, Laugwitz was the first to suggest both the German “reziproke Kettenwurzel” and the English “continued reciprocal root.”

  3. 3.

    Such constructions had been proposed as early as 1924 [1, 12, 20, 32], although Laugwitz does not mention them.

  4. 4.

    Anticipated some 20 years earlier by Kakeya [20], the \(f\)-expansion was introduced in 1944 by Bissinger [1] as a generalization of the continued fraction expansion. Though notationally identical to (7), the terms of an \(f\)-expansion are assumed to be integers.

  5. 5.

    On the other hand, the references given here regarding continued radicals are a small sampling from an extensive literature. The online document [19] lists many other works on continued radicals from the nineteenth and early twentieth centuries.

  6. 6.

    The “left-aligned exponent” notation, as well as the English term “continued power,” were introduced by Dixon [9] in 1878.

References

  1. 1.

    Bissinger, B.H.: A generalization of continued fractions. Bull. Am. Math. Soc. 50, 868–876 (1944)

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    Bochow, K.: Kettenwurzeln und Winkelfunktionen. Z. Math. Nat. Unterr. 41, 161–186 (1910)

    Google Scholar 

  3. 3.

    Bouché, A.: Premier essai sur la théorie des radicaux continus, et sur ses applications à l’algébre et au calcul infinitésimal. Mém. Soc. Acad. Maine et Loire. 12, 81–151 (1862)

    Google Scholar 

  4. 4.

    Brezinski, C.: History of Continued Fractions and Padé Approximations. Springer Series in Computational Mathematics, 12. Springer, Berlin (1991)

    Google Scholar 

  5. 5.

    Candido, G.: Sul numero \(\pi \). Suppl. Period. Mat. 11, 113–115 (1908)

    MATH  Google Scholar 

  6. 6.

    Réalis, S.: Sur quelques questions proposées dans la nouvelle correspondance, Question 142. Nouv. Corresp. Math. 3, 193–194 (1877)

  7. 7.

    Cipolla, M.: Intorno ad un radicale continuo. Period. mat. l’insegnamento second., Ser. 3. 5, 179–185 (1908)

    MATH  Google Scholar 

  8. 8.

    Cwojdzinski, K.: Kettenwurzeln. Arch. Math. Phys. Ser. 2. 17, 29–35 (1899)

  9. 9.

    Dixon, T.S.E.: Continued roots. The Analyst 5, 20–21 (1878)

    Article  Google Scholar 

  10. 10.

    Doppler, C.: Über Kettenwurzeln und deren Konvergenz. Jahrb. Polytech. Inst. Wien 17, 175–200 (1832)

    Google Scholar 

  11. 11.

    Eichhorn, C. F.: Principien einer allgemeinen Functionenrechnung. Helwingschen Hof-Buchhandlung, Hannover, pp. 81–98 and 138–139 (1834).

  12. 12.

    Everett, C.I.: Representations for real numbers. Bull. Am. Math. Soc. 52, 861–869 (1946)

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    Günther, S.: Eine didaktisch wichtige Lösung der trinomischen Gleichung. Hoffmann Z. 11, 68–72 (1880)

    Google Scholar 

  14. 14.

    Herschfeld, A.: On infinite radicals. Am. Math. Mon. 42, 419–429 (1935)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hoffmann, K.E.: Ueber die Auflösung der trinomischen Gleichungen durch kettenbruchähnliche Algorithmen. Arch. Math. Phys. 66, 33–45 (1881)

    Google Scholar 

  16. 16.

    Isenkrahe, C.: Über die Anwendung iteriter Funktionen zur Darstellung der Wurzeln algebraischer und transcendenter Gleichungen. Math. Ann. 31, 309–317 (1888)

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Jones, D.J.: Continued powers and roots. Fibonacci Q. 9, 37–46 (1991)

    Google Scholar 

  18. 18.

    Jones, D.J.: Continued powers and a sufficient condition for their convergence. Math. Mag. 68, 387–392 (1995)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Jones, D. J.: A chronology of continued radicals and other selected chain compositions. https://www.academia.edu/7564095/A_chronology_of_continued_radicals_and_other_selected_chain_compositions_2014-07-05 (2014) Accessed 5 July 2014

  20. 20.

    Kakeya, S.: On a generalized scale of notation. Jpn. J. Math. 1, 95–108 (1924)

    Google Scholar 

  21. 21.

    Laugwitz, D.: Kettenwurzeln und Kettenoperationen. Elem. Math. 45, 89–98 (1990)

    MATH  MathSciNet  Google Scholar 

  22. 22.

    Laugwitz, D., Schönefuss, L.: Convergence of continued operations. Proc. Eur. Conf. Iteration Theory. Grazer Math. Ber. 339, 243–250 (1999)

    MATH  Google Scholar 

  23. 23.

    Lucas, E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1, 184–240 (1878)

    MATH  Article  Google Scholar 

  24. 24.

    Martin, G.: The unreasonable effectualness of continued function expansions. J. Aust. Math. Soc. 77, 305–319 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    M. R. S.: Note sur quelques expressions algébriques peu connues. Ann. Math. Pure. Appl. 20, 352–366 (1830)

  26. 26.

    Pólya, G.: Problem proposal. Arch. Math. Phys. Ser. 3 24, 84 (1916)

    Google Scholar 

  27. 27.

    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis, vol. 1. Springer, Berlin (1925)

    Google Scholar 

  28. 28.

    Ramanujan, S.: Question 289. J. Indian Math. Soc. 3, 90 (1911)

    Google Scholar 

  29. 29.

    Schönefuss, L. W.: Nichtautonome Differenzengleichungen und Kettenoperationen. Mitteilungen aus dem Mathem. Seminar Giessen 207 (1992).

  30. 30.

    Schmidten, H.G.: Mémoire sur l’intégration des équations linéaires. Ann. Math. Pure Appl. 11, 269–316 (1821)

    Google Scholar 

  31. 31.

    Sizer, W.S.: Continued roots. Math. Mag. 59, 23–27 (1986)

    MATH  MathSciNet  Article  Google Scholar 

  32. 32.

    Thron, W.J.: Convergence regions for continued fractions and other infinite processes. Am. Math. Mon. 68, 734–750 (1961)

    MATH  MathSciNet  Article  Google Scholar 

  33. 33.

    Viète, F.: Variorum de Rebus Mathematicis Responsorum, Liber VII (1593) [Facsimile excerpts reproduced in Berggren, L., Borwein, J., Borwein, P.: Pi, A Sourcebook, pp. 53–67. Springer, New York (1997)]

  34. 34.

    Wiernsberger, P.: Recherches diverses sur des polygones réguliers et les radicaux superposés. A. Rey, Lyons (1904)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dixon J. Jones.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jones, D.J. Continued reciprocal roots. Ramanujan J 38, 435–454 (2015). https://doi.org/10.1007/s11139-014-9594-3

Download citation

Keywords

  • Continued power
  • Continued reciprocal root
  • Continued radical
  • Continued root
  • Continued fraction

Mathematics Subject Classification

  • 40A99
  • 40A05