Short, Ian
(2006).
A counterexample to a continued fraction conjecture.
Proceedings of the Edinburgh Mathematical Society. Series II, 49(3) pp. 735–737.
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Abstract
It is known that if a ∈ ℂ \(−∞,−¼) and a_{n} → a as n → ∞, then the infinite continued fraction with coefficients a_{1}, a_{2},... 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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