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

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