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Continued reciprocal roots


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 \(q=|1/p|\).

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

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

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


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Correspondence to Dixon J. Jones.

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Jones, D.J. Continued reciprocal roots. Ramanujan J 38, 435–454 (2015).

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

Mathematics Subject Classification

  • 40A99
  • 40A05