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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)

## Contents |

## 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

- Weisstein, Eric W., Khinchin's Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/KhinchinsConstant.html]
- Weisstein, Eric W., Continued Fraction Constants, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/ContinuedFractionConstants.html]