In the analytic theory of continued fractions, a chain sequence is an infinite sequence {a<sub>n</sub>} of non-negative real numbers chained together with another sequence {g<sub>n</sub>} of non-negative real numbers by the equations
where either (a) 0 ≤ g<sub>n</sub> < 1, or (b) 0 < g<sub>n</sub> ≤ 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 {a<sub>n</sub>} are a chain sequence.
The sequence , , , ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g<sub>0</sub> = g<sub>1</sub> = g<sub>2</sub> = ... = , it is clearly a chain sequence. This sequence has two important properties.