Some periodic sequences of circular convergence regions

Further results on convergence acceleration for continued fractions 𝐾(𝑎_{𝑛}/1)

“…By Example 3.3 we know that |n" + f(n)' \> 8 > 0 from some « on, and that a'p = 0 for one or more p G {1,2,..., A} is permissible. This is an example where also [7,Theorem 4.1] applies, but, as will be shown in §5, with not too much advantage. The following numerical example illustrates this.…”

“…This theorem states a kind of generalization of an earlier result by the author [7,Theorem 4.1], since it is valid for a much larger selection of continued fractions; larger where both the admissible auxiliary continued fractions and the possible speed of convergence of | an -a'n \ are concerned. In many cases it also gives better bounds than [7] does, for \f -f* \/\f -fn \. This is shown in §5.…”

“…The inequality (3.8) may seem rather rough. However, in [7] it is proved that under mild conditions we have…”

“…In §2 some basic definitions and notation are given, in §4 some examples, and in §5 the main theorem of this paper is compared to earlier results in [7,14]. Further, the ordinary and the modified approximants can be written (2.5) fn = S"(0) = ^, f: = Sn(w") for « = 0,1,2,..., respectively, and (2.6) «" = -A-= -SB-'(oo) for« =1,2,3,..., Ä»-i where «"is defined by (1.9), and…”

“…Their work also includes estimates for the ratio between the truncation errors for the ordinary approximant, /", and the modified one,/,*, and thus an estimate for the improvement obtained by this method. In [7], the author extended part of their theory to more general continued fractions. This time, vv" in (1.5) is chosen to be the finite tail/*"' of some known auxiliary continued fraction K(a'n/\) where an -a'n^> 0.…”