词条 | Chain sequence |
释义 |
In the analytic theory of continued fractions, a chain sequence is an infinite sequence {an} of non-negative real numbers chained together with another sequence {gn} of non-negative real numbers by the equations where either (a) 0 ≤ gn < 1, or (b) 0 < gn ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite continued fractions. The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem[1] shows that converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {an} are a chain sequence. An exampleThe sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = ½, it is clearly a chain sequence. This sequence has two important properties.
generates the same unending sequence {¼, ¼, ¼, ...}. Notes1. ^Wall traces this result back to Oskar Perron (Wall, 1948, p. 48). References
1 : Continued fractions |
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