The hyperbolic geometry of continued fractions K(1|bn)

Short, Ian (2006). The hyperbolic geometry of continued fractions K(1|bn). Annales Academiae Scientiarum Fennicae Mathematica, 31(2) pp. 315–327.

URL: https://www.acadsci.fi/mathematica/Vol31/short.htm...

Abstract

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

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