A Theorem on the Convergence of a Continued Fraction Page: 2
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2
the spproximsnta of the continued fraction
a
1
bi + !s.
b2
+ .
If Pt most a finite number of vanish *nd the sequence ^ ha#
a finite lia&t teen the continued fraction is said to converge
to this limit ©a? to he convergent
Sine# a continued fraction may he considered as a sequone©
of linear fractional transformation® we will prove some gen-
eral theorems of linear fractional transformations•
theorem l.l~~Xf |z - %\± ft and w * bZ b^0# then there
exists a number c and a real positive number r such that
w - e|f r. Also if |w - e|£ r, then |z * q^iz 1.
Let c bQ and r * 8 |h[ • Since w * bZ, then Z —? *
If |Z - Q|f then _ %|f 1, or jw - bQj^ ft|b|t or
jw - c| f r. therefore w Is in a circle with cent or c * bQ and
radius r 38 S Jb]
If [w * e| £ r, then since w « bZ, c * Qb, and r * 1 lb I we
have jbZ - Qb|f t ]fe| # or |z - q|^l R.
^H. S. Wall, Continued Fractions. p. 16.
2Ibid., p. 19.
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Kostelec, John C. A Theorem on the Convergence of a Continued Fraction, thesis, January 1953; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc107838/m1/5/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .