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Convergents constant

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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

  1. Express with the convergents of as a continued fraction
  2. Compute the convergents
    (using the efficient recursive method shown on generalized continued fractions convergents;)
  3. The next iterate is

For example, starting with , where is a positive real constant, first obtain the simple continued fraction

giving convergents


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