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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Convergence acceleration for continued fractions $ K(a\sb{n}/1)$


Author: Lisa Jacobsen
Journal: Trans. Amer. Math. Soc. 275 (1983), 265-285
MSC: Primary 40A15; Secondary 30B70
DOI: https://doi.org/10.1090/S0002-9947-1983-0678349-1
MathSciNet review: 678349
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Abstract: A known method for convergence acceleration of limit periodic continued fractions $ K({a_n}/1),{a_n} \to a$, is to replace the approximants $ {S_n}(0)$ by "modified approximants" $ {S_n}({f^{\ast}})$, where $ f^{\ast} = K(a/1)$. The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be $ {S_n}({f^{(n)^{\prime}}})$, where $ K({a^{\prime}_n}/1)$ is a converging continued fraction whose tails $ {f^{(n)\prime}}$ are all known, and where $ {a_n} - a_n^\prime \to 0$.

As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678349-1
Keywords: Continued fractions, convergence acceleration, uniform convergence regions
Article copyright: © Copyright 1983 American Mathematical Society