Let A = {an}∞n = 1 be a sequence of positive integers. There are two related sequences Pn and Qn obtained from A by taking partial convergents out of the number [0; a1, a2, ..., an, ...], where Pn and Qn are the numerators and denominators of the finite continued fraction [0; a1, a2, ...,an].
Let P(n) be the largest positive integer k , such that Pk ≤ n. The sequence Q(n) is defined similarly. • A known result of Barnes' Theorem states that
P (
n ) =
o (
n ) and
Q (
n ) =
o (
n ). • In this paper we improve this result as
P (
n ) =
O (log n) and
Q (
n ) =
O (log n), where it follows that
P (
n )=
o (
nε ) and
Q (
n ) =
o (
nε ) for any
ε >0.
| Identifer | oai:union.ndltd.org:unf.edu/oai:digitalcommons.unf.edu:etd-1085 |
| Date | 01 January 1994 |
| Creators | Vafabakhsh, Seyed J |
| Publisher | UNF Digital Commons |
| Source Sets | University of North Florida |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | UNF Theses and Dissertations |
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