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

Jacobsen, Lisa

doi:10.1090/S0002-9947-1984-0719662-X

Further results on convergence acceleration for continued fractions $K(a_{n}/1)$
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Trans. Amer. Math. Soc. 281 (1984), 129-146 Request permission

Abstract:

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

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