A Theorem on the Convergence of a Continued Fraction Page: 5
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or !Z+lH+«Z+«Ht (m)4 2p,
or WZ + al+ 2^ (IB) t2^ (Si) 4 2p,
gnnn
or t&+ aZ"f:2p
1,7""It \Z - q|^ B and w =
th®n
(1) If l<i|- R * © then thar® exists a number m/o and a
real number <j such that 1W4 *5 ~2q,. Furthermore if
wtr4 ftw 52q . then | Z -
(ii) if |q| -■ R^O, than there exists a number c and a real
number r such that \w - ©j£ r Also if - ©|f: r*
than \z * q|"5 B,
(iii) if lQ| - S^D, than there exi#ts a number c «nd A raal
number r aueh that \w - a(z r. Also if \w - ©/z. r,
than \Z - Qjf R.
Since w « 1 than Z « • • If |z - Qlf- Bt
z i 1 . , * ' 1
than \ y - Qj'5 S;
|l - «q|* B ^ w|;
[l - (j - i$] 2 B8 |w| 1
L, ci
*
tf((& • **) - *Q - ml - X- (1.1)
(i) If |ft| - R a % than |Q| * i# or jof 12# and |qf - 12 « ©•
I/at • = -$, *«-Q and t « - l/2.
fhus from aquation (1*1) wa - • (Hr - X.
Hanca if Z is in the circle jz - Ql^ R, then w ia in the
half plan® * 4 aw f2q.
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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/m1/8/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .