If you use all the convergents of the simple continued fraction of a positive real constant as the terms of a generalized continued fraction, then likewise use the new convergents in another generalized continued fraction, and so on... ad infinitum, for all numbers in the unit interval , you get the convergents constant for all numbers of that interval. (Cf. Talk:Table_of_convergents_constants#Open_Problem)
Iterated continued fractions from convergents
Cf. Iterated continued fractions from convergents.
In order to get from iterate to iterate
- Express with the convergents of as a continued fraction
- Compute the convergents
(using the efficient recursive method shown on generalized continued fractions convergents;)
- The next iterate is
For example, starting with , where is a positive real constant, first obtain the simple continued fraction
Convergence of iterated continued fractions from convergents
We define the limit of iterated continued fractions from convergents for a constant as
Table of convergents constants
Cf. Table of convergents constants.