Chain sequence


In the analytic theory of continued fractions, a chain sequence is an infinite sequence of non-negative real numbers chained together with another sequence of non-negative real numbers by the equations
where either 0 ≤ gn < 1, or 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 are a chain sequence.

An example

The sequence 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.