A Theorem on the Convergence of a Continued Fraction Page: 11
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11
^Theorem 1.9—If w = | end aZ 12 ^2p, */<> th®a
(1) if p s 0, thara exists a number b^O and a r@®l mimfoar
q suefe that b®r-f tiff—8%. Furthermore if bw4tir~2q,
than 1Z+ aZ~2p.
Cil) if p>0, then there ©xists a number c and a real ouabor
j> such th&t |v ** cj^ r • Also if jw — c|^. r* than
fZ+iS^gp.
(Hi) if p<0, then there exists a number c and a raal nwntew
r such that - c^r r* Also if - e/— r, than.
aZ-f aZ—2p*
Sinca w = | , than Z = | « 2p 2 * .
If IZ-f aZ ^ 2p#
thon — -f •y-Tgr-
/w/ /w/
SW-^aw £ 2pww (1.5)
(i) if p « 0, Let p = q a = b, and a b. Thus from
aquation (1.5) w® gat aw SW^O. Thus w is in tha half
plana bw + bw .£ 2q a 0.
Furthermore if bw tiff 2q, than sine© q = p 0,
b * a and b « a w© have
aw + m-2p = 0 = 2p/w/2 ,
or au,,lS,4- a—JL—— ~ 2p.
>/* H8 _
Since Z = „ ■ y., , then 1Z + Z—Sp.
fir
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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/m1/14/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .