Improved truncation error bounds for limit periodic continued fractions with additional assumptions on its elements

摘要

It is well known that for convergent pure periodic continued fractions (i.e.: |r|=|x1/x2|<1) the truncation error is O(|r|n). In earlier articles, with assumptions weaker than ∑m=1∞mdm<∞, it was shown that the truncation error for limit periodic continued fractions is, at best, of the form K(|r′|)|r′|n,0<|r′|<|r|, where K(|r′|) is a function of |r′| which may tend to infinity as |r′|→|r|. It is thus of interest to determine whether there exist conditions on {an},{bn} in the limit periodic continued fraction K(an/bn), which would insure that the truncation error is O(|r|n). It is shown here that restrictions of that kind do exist and that ∑ndn<∞ is such a condition. In view of the result on pure periodic continued fractions, mentioned above, the estimate here obtained is optimal. Whether, beyond its aesthetic appeal, this optimal error bound might also be useful, is not known to the author.