On a certain transformation of continued fractions

Tales about tails

“…Tails of continued fractions (also wrong tails) are of vital importance for modifications of continued franctions. The survey article [13] discusses modifications and their applications, and also contains an extensive bibliography. Newer results are found in [4][5][6][7].…”

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

“…Possibly, better results can be found by using properties of periodic auxiliary continued fractions more extensively. A comparison with the result for k = 1 proved by Thron and Waadeland [14], shows, however, that Theorem 3.1 is compatible. (See §5.)…”

“…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…”

“…In [14], Thron and Waadeland developed a complete theory for using these modified approximants in the limit 1-periodic case with a^O. 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.…”

“…The special case k-\, that is a'n = -\, was treated in detail in [14]. For comparison between the two results, see §5.…”