Chain Sequence (continued Fraction)
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 shows that
converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {an} are a chain sequence.
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