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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
giving convergents
then
giving convergents
giving convergents
Convergence of iterated continued fractions from convergents
We define the limit of iterated continued fractions from convergents for a constant as
Convergents constants
Table of convergents constants
Cf. Table of convergents constants.
See also
External links