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Monatshefte für Mathematik

, Volume 111, Issue 2, pp 119–126 | Cite as

Continued fractions for some alternating series

  • J. L. Davison
  • J. O. Shallit
Article

Abstract

We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant
$$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$
is transcendental. Here (S n ) isSylvester's sequence defined byS0=2 andS n+1 =S n 2 S n +1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.

Keywords

Continue Fraction Partial Quotient Transcendental Number 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. L. Davison
    • 1
  • J. O. Shallit
    • 2
  1. 1.Mathematics and Computer ScienceLaurentian UniversitySudburyCanada
  2. 2.Computer ScienceUniversity of WaterlooWaterlooCanada

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