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

Abstract

If K ( a n ′ / 1 ) K(a_n’/1) is a convergent continued fraction with known tails, it can be used to construct modified approximants f n ∗ f_n^{\ast } for other continued fractions K ( a n / 1 ) K({a_n}/1) with unknown values. These modified approximants may converge faster to the value f f of K ( a n / 1 ) K({a_n}/1) than the ordinary approximants f n {f_n} do. In particular, if a n − a n ′ → 0 {a_n} - a_n’ \to 0 fast enough, we obtain | f − f n ∗ | / | f − f n | → 0 |f - f_n^{\ast }|/|f - {f_n}| \to 0 ; i.e. convergence acceleration. the present paper also gives bounds for this ratio of the two truncation errors, in many cases.