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 K(n=1)(infinity)a(n)/1 converge to different values, then lim(n-->infinity) \a(n)\ = infinity. (C) 2004 Elsevier B.V. All rights reserved.

[发布日期] 2004-12-01

[关键词] continued fractions;general convergence;q-continued fraction;Rogers-Ramanujan