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|DOI (Digital Object Identifier) Link:||http://doi.org/10.1017/S0013091505000581|
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It is known that if a ∈ ℂ \(−∞,−¼) and an → 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 an → 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.
|Item Type:||Journal Article|
|Copyright Holders:||2006 Edinburgh Mathematical Society|
|Keywords:||Primary 40A15; 30B70; continued fractions; Möbius transformations; iteration; dynamics|
|Academic Unit/Department:||Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
|Depositing User:||Ian Short|
|Date Deposited:||30 Jul 2010 14:03|
|Last Modified:||03 Aug 2016 16:19|
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