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 ≤ g_n < 1, or 0 < g_n ≤ 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 = ... =, it is clearly a chain sequence. This sequence has two important properties.

Since f = x − x² is a maximum when x =, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if =, and x <, the resulting sequence will be an endless repetition of a real number y that is less than.
The choice g_n = is not the only set of generators for this particular chain sequence. Notice that setting