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The closure of convergence sets for continued fractions are convergence sets

Published online by Cambridge University Press:  20 January 2009

Lisa Lorentzen
Lisa Lorentzen
Affiliation:
Division of Mathematical Sciences, University of Trondheim, NTH, N-7034 Trondheim, Norway
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Abstract

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We prove that if Ω is a simple convergence set for continued fractions K(an/bn), then the closure of Ω is also such a convergence set. Actually, we prove more: every continued fraction K(an/bn) has a “neighbourhood” where rn>0 and sn>0, with the following property: Every continued fraction from {n} converges if and only if K(an/bn) converges.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 1 , February 1994 , pp. 39 - 46
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Atkinson, F. V., Discrete and Continuous Boundary Problems (Academic Press, New York, 1964).Google Scholar
2.Jacobsen, L., Nearness of continued fractions, Math. Scand. 60 (1987), 129–147.CrossRefGoogle Scholar
3.Jacobsen, L. and Waadeland, H., Some useful formulas involving tails of continued fractions (Lecture Notes in Math. 932, Springer, Berlin, 1982), 99–105.CrossRefGoogle Scholar
4.Jones, W. B. and Thron, W. J., Continued Fractions. Analytic Theory and Applications (Addison Wesley, Encyclopedia of Mathematics and its Applications, 1980).Google Scholar
5.Lorentzen, L., Analytic continuation of functions given by continued fractions, revisited, Rocky Mountain J. of Math. 23 (1993), 683–706.CrossRefGoogle Scholar
6.Lorentzen, L., Divergence of continued fractions related to hypergeometric series, Math. Comp., to appear.Google Scholar
7.Njåstad, O., A survey of some results on separate convergence of continued fractions (Lecture Notes in Math. 1406, Springer-Verlag), 88–115.Google Scholar
8.Waadeland, H., Tales about tails, Proc. Amer. Math. Soc. 90 (1984), 57–64.CrossRefGoogle Scholar