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arXiv · 1812.10878

A Theorem on Divergence in the General Sense for Continued Fractions

Douglas Bowman·James Mc Laughlin
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Abstract

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of $q$ continued fraction to show, that if $G(q)$ is one of these continued fractions and $|q|>1$, then either $G(q)$ converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction $K_{n=1}^{\infty}a_{n}/1$ converge to different values, then $\lim_{n \to \infty}|a_{n}| = \infty$.

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Douglas Bowman, James Mc Laughlin. 2018-12-28. A Theorem on Divergence in the General Sense for Continued Fractions. https://doi.org/10.1016/j.cam.2004.02.012

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