A counterexample to a continued fraction conjecture

Short, Ian (2006). A counterexample to a continued fraction conjecture. Proceedings of the Edinburgh Mathematical Society. Series II, 49(3) pp. 735–737.

DOI: https://doi.org/10.1017/S0013091505000581

Abstract

It is known that if a ∈ ℂ \(−∞,−¼) and a_n → a as n → ∞, then the infinite continued fraction with coefficients a₁, a₂,... converges. A conjecture has been recorded by Jacobsen et al., taken from the unorganized portions of Ramanujan’s notebooks, that if a ∈ (−∞,−¼) and a_n → a as n→∞, then the continued fraction diverges. Counterexamples to this conjecture for each value of a in (−∞,−¼) are provided. Such counterexamples have already been constructed by Glutsyuk, but the examples given here are significantly shorter and simpler.

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