, Volume 111, Issue 2, pp 119-126

Continued fractions for some alternating series

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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 byS 0=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.

Research supported in part by NSF grant CCR-8817400 and a Walter Burke award from Dartmouth College.