Abstract
A previous divergence criterion by A. Magnus and the author is generalized. We show that K(an/1) diverges (by oscillation) if an=−1/4−(c+εn)/n(n+1), where c>1/16 and εn ε ℝ, |ε|=O(1/logαn), α>2.
Keywords
- Cauchy Sequence
- Continue Fraction
- Convergence Region
- Linear Fractional Transformation
- Continue Fraction Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Gill, J., Infinite Compositions of Möbius Transformations, Trans. Amer. Math. Soc. 176 (1973), 479–487.
Jacobsen, L., Composition of Linear Fractional Transformations in Terms of Tail Sequences, to appear in Proc. Amer. Math. Soc.
Jacobsen, L., Nearness of Continued Fractions, to appear in Math Scand.
Jacobsen, L. and Magnus, A., On the Convergence of Limit Periodic Continued Fractions K(an/1), where an → −1/4, Lecture Notes in Math. 1105 (1984), 243–248.
Jacobsen, L. and Waadeland, H., Even and Odd Parts of Limit Periodic Continued Fractions, to appear in J. CAM.
Jones, W. B. and Thron, W. J., Continued Fractions: Analytic Theory and Application, Encyclopedia of Math. and Its Appl., Addison-Wesley, 1980.
Perron, O., Die Lehre von den Ketterbrüchen, Band II, B. G. Teubner, 1957.
Pringsheim, I., Über die Konvergenz unendlicher Kettenbrüche, S.-B. Bayer. Akad. Wiss. Math.-Mat. Kl. 28 (1899), 295–324.
Thron, W. J., On Parabolic Convergence Regions for Continued Fractions, Math. Zeitschr. 69 (1958), 173–182.
Thron, W. J. and Waadeland, H., Accelerating Convergence of Limit Periodic Continued Fractions K(an/1), Numer. Math. 34 (1980), 155–170.
Waadeland, H., Tales about Tails, Proc. Amer. Math. Soc. 90 (1984), 57–64.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag
About this paper
Cite this paper
Jacobsen, L. (1986). On the convergence of limit periodic continued fractions K(an/1), where an→−1/4. Part II. In: Thron, W.J. (eds) Analytic Theory of Continued Fractions II. Lecture Notes in Mathematics, vol 1199. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075934
Download citation
DOI: https://doi.org/10.1007/BFb0075934
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16768-6
Online ISBN: 978-3-540-38817-3
eBook Packages: Springer Book Archive
