# Chain sequence

In the analytic theory of continued fractions, a **chain sequence** is an infinite sequence {*a*_{n}} of non-negative real numbers chained together with another sequence {*g*_{n}} of non-negative real numbers by the equations

where either (a) 0 ≤ *g*_{n} < 1, or (b) 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[1] shows that

converges uniformly on the closed unit disk |*z*| ≤ 1 if the coefficients {*a*_{n}} are a chain sequence.

## An example

The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting *g*_{0} = *g*_{1} = *g*_{2} = ... = ½, it is clearly a chain sequence. This sequence has two important properties.

- Since
*f*(*x*) =*x*−*x*^{2}is a maximum when*x*= ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {*g*_{n}} = {*x*}, and*x*< ½, the resulting sequence {*a*_{n}} will be an endless repetition of a real number*y*that is less than ¼. - The choice
*g*_{n}= ½ is not the only set of generators for this particular chain sequence. Notice that setting

- generates the same unending sequence {¼, ¼, ¼, ...}.

## Notes

- Wall traces this result back to Oskar Perron (Wall, 1948, p. 48).

## References

- H. S. Wall,
*Analytic Theory of Continued Fractions*, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), ISBN 0-8284-0207-8