A Theorem on the Convergence of a Continued Fraction Page: 13
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IS
Theorem jz - oj^M and If * s ^AtJfL * where
oC - 2fB ^ 0, then
Ci) If £ a 0, than there exists a number c and a real
xxumber r such that r« Also if jw - c/— r,
then ^2 - Q^£r«,
(ii) if 0, then
a) if y + / /^/ % s 0, then tiief© exists a
number a^t 0 pnd a re*! number p such that
¥w-faW^2p, Also if lto+8p then jz ~ o|-H,
then there exists a
number e sad a real number r such that Jw - ej —
Also if jw - cjS: r, then jz - Q,J — H,
c) if Is 0, then there exist# a
number c and a real number r such that r
Also if /w - cj^r, then )z - R.
Ci) If 2T® 0# then w <= c£ Z ~h-^r- * W*| *® Z clziicI
w = «JL Since jz ** <^^R, then by Theorem 1*1
there is a and an r^ auch that c^r^. By
Theorem 1.4 there is a c and an r auch that - c^— r«
If |sr - c|^. r, then jz - R my be proved by the
.same method of reasoning as above.
(ii) {• If £fj£0* Let *g — Wj =s ¥g+ iT<T * *4 s l/*3
*s m~*urht4§ and w = wg ...^« Since Jz - <^i=R, then by
Theorem 1.1 there Is a Cg and an rg auch that ^Wg - ■© r2
where eg ~ eind rg |^|2R. By Theorem 1.4 there it a
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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/m1/16/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .