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The hyperbolic geometry of continued fractions <b>K</b>(1|<i>b<sub>n</sub></i>)

By Ian Short

Abstract

The Stern-Stolz theorem states that if the infinite series ∑|<i>b<sub>n</sub></i>| converges, then the continued fraction <b>K</b>(1|<i>b<sub>n</sub></i>) diverges. H. S. Wall asks whether just convergence, rather than absolute convergence of ∑<i>b<sub>n</sub></i> is sufficient for the divergence of <b>K</b>(1|<i>b<sub>n</sub></i>). \ud We investigate the relationship between ∑|<i>b<sub>n</sub></i>| and <b>K</b>(1|<i>b<sub>n</sub></i>) with hyperbolic geometry and use this geometry to construct a sequence <i>b<sub>n</sub></i> of real numbers for which both ∑|<i>b<sub>n</sub></i>| and \ud <b>K</b>(1|<i>b<sub>n</sub></i>) converge, thereby answering Wall's question

Year: 2006
OAI identifier: oai:oro.open.ac.uk:22452
Provided by: Open Research Online

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