A theorem on divergence in the general sense for continued fractions

摘要

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 Kn=1∞an/1 converge to different values, then limn→∞|an|=∞.