A Theorem on the Convergence of a Continued Fraction Page: 9
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Theorem \Z and w « g
t
(i) If lQ| - 8 = C, than there exists a nomber a ^ 0 and &
real number f such that 1% aW 8fc# Fwthewnofce If
awftSrfat* then \Z - <4\Z1B.
(11) if lQ| • R >0, then there exista a number e and & real
number r aueh that |w - el£r. Also if \v - e\^. r# then
\Z • R.
(Ill )lf |Q| - 1 ^.0, then there exists a number c nd a real
number r aueh that |w - ©J £ r. Also If \w - c)^; r# then
\z - a|> i.
Since w * | , mm Z | . If (1 - Q|> t then
If *" ^1—fi* Thm by the sane method ©# used In Theorem 3,1
w© get nf ( Q 2 - ®*} - I? - wQZ: *1. _ (l t)
(1) If \Q\ ~ a = 0, then |q| 2- R2 0. Let a = 1#
* « Q, end % l/2. Then from equation (1.2) we have
- Qw - XSt Z> 1
+ Sr 1
Sw+- «w f; 2%
Henee if Z ia in the region |Z - then w is in the half
plane rf
If aw+ a¥ ^ 8<| then since a = "5, a <*, and q * i/2#
we have Qr-fCH1, ^ 1,
or - Qk - Since |Q\8 - R2 « 0, then irw( |q| 2 - B2)
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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/m1/12/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .