Convergents constant

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If you use all the convergents of the simple continued fraction of a positive real constant $x \in (n, n + 1)$ 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 $(n, n + 1)$ , you get the convergents constant for all numbers of that interval. (Cf. Talk:Table_of_convergents_constants#Open_Problem)

Iterated continued fractions from convergents

In order to get from iterate $x_{n}$ to iterate $x_{n + 1}$

Express $x_{n}$ with the convergents $c_{i} (n - 1)$ of $x_{n - 1}$ as a continued fraction $[c_{0} (n); c_{1} (n), c_{2} (n), \dots] = \frac{1}{q_{0} (n)} [p_{0} (n); q_{0} (n) q_{1} (n) / p_{1} (n), q_{1} (n) q_{2} (n) / p_{2} (n), . . .];$
Compute the convergents $c_{0} (n + 1) \equiv \frac{p_{0} (n + 1)}{q_{0} (n + 1)} = \frac{1}{q_{0} (n)} (p_{0} (n)), c_{1} (n + 1) \equiv \frac{p_{1} (n + 1)}{q_{1} (n + 1)} = \frac{1}{q_{0} (n)} (p_{0} (n) + \frac{q_{0} (n) q_{1} (n)}{p_{1} (n)}), c_{2} (n + 1) \equiv \frac{p_{2} (n + 1)}{q_{2} (n + 1)} = \frac{1}{q_{0} (n)} (p_{0} (n) + \frac{q_{0} (n) q_{1} (n)}{p_{1} (n) + \frac{q_{1} (n) q_{2} (n)}{p_{2} (n)}}), \dots$
(using the efficient recursive method shown on generalized continued fractions convergents;)
The next iterate is $x_{n + 1} = [c_{0} (n + 1); c_{1} (n + 1), c_{2} (n + 1), \dots] .$

For example, starting with $x_{0} \equiv x$ , where $x$ is a positive real constant, first obtain the simple continued fraction

x_{0} = [a_{0} (0); a_{1} (0), a_{2} (0), \dots] = a_{0} (0) + \frac{1}{a_{1} (0) + \frac{1}{a_{2} (0) + \frac{1}{a_{3} (0) + \frac{1}{a_{4} (0) + \frac{1}{a_{5} (0) + \frac{1}{a_{6} (0) + \frac{1}{a_{7} (0) + \frac{1}{⋱}}}}}}}},

giving convergents

{\frac{p_{0} (1)}{q_{0} (1)}, \frac{p_{1} (1)}{q_{1} (1)}, \frac{p_{2} (1)}{q_{2} (1)}, \frac{p_{3} (1)}{q_{3} (1)}, \frac{p_{4} (1)}{q_{4} (1)}, \frac{p_{5} (1)}{q_{5} (1)}, \frac{p_{6} (1)}{q_{6} (1)}, \frac{p_{7} (1)}{q_{7} (1)}, . . .}

then

x_{1} = \frac{p_{0} (1)}{q_{0} (1)} + \frac{1}{\frac{p_{1} (1)}{q_{1} (1)} + \frac{1}{\frac{p_{2} (1)}{q_{2} (1)} + \frac{1}{\frac{p_{3} (1)}{q_{3} (1)} + \frac{1}{\frac{p_{4} (1)}{q_{4} (1)} + \frac{1}{\frac{p_{5} (1)}{q_{5} (1)} + \frac{1}{\frac{p_{6} (1)}{q_{6} (1)} + \frac{1}{\frac{}{} + \frac{1}{⋱}}}}}}}}

= \frac{1}{q_{0} (1)} {p_{0} (1) + \frac{q_{0} (1) q_{1} (1)}{p_{1} (1) + \frac{q_{1} (1) q_{2} (1)}{p_{2} (1) + \frac{q_{2} (1) q_{3} (1)}{p_{3} (1) + \frac{q_{3} (1) q_{4} (1)}{p_{4} (1) + \frac{q_{4} (1) q_{5} (1)}{p_{5} (1) + \frac{q_{5} (1) q_{6} (1)}{p_{6} (1) + \frac{q_{6} (1) q_{7} (1)}{p_{7} (1) + \frac{}{}}}}}}}}}

= \frac{1}{q_{0} (1)} {a_{0} (1) + \frac{b_{1} (1)}{a_{1} (1) + \frac{b_{2} (1)}{a_{2} (1) + \frac{b_{3} (1)}{a_{3} (1) + \frac{b_{4} (1)}{a_{4} (1) + \frac{b_{5} (1)}{a_{5} (1) + \frac{b_{6} (1)}{a_{6} (1) + \frac{b_{7} (1)}{a_{7} (1) + \frac{}{}}}}}}}}},

giving convergents

{\frac{p_{0} (2)}{q_{0} (2)}, \frac{p_{1} (2)}{q_{1} (2)}, \frac{p_{2} (2)}{q_{2} (2)}, \frac{p_{3} (2)}{q_{3} (2)}, \frac{p_{4} (2)}{q_{4} (2)}, \frac{p_{5} (2)}{q_{5} (2)}, \frac{p_{6} (2)}{q_{6} (2)}, \frac{p_{7} (2)}{q_{7} (2)}, . . .}

$\dots$

x_{n} = \frac{p_{0} (n)}{q_{0} (n)} + \frac{1}{\frac{p_{1} (n)}{q_{1} (n)} + \frac{1}{\frac{p_{2} (n)}{q_{2} (n)} + \frac{1}{\frac{p_{3} (n)}{q_{3} (n)} + \frac{1}{\frac{p_{4} (n)}{q_{4} (n)} + \frac{1}{\frac{p_{5} (n)}{q_{5} (n)} + \frac{1}{\frac{p_{6} (n)}{q_{6} (n)} + \frac{1}{\frac{}{} + \frac{1}{⋱}}}}}}}}

= \frac{1}{q_{0} (n)} {p_{0} (n) + \frac{q_{0} (n) q_{1} (n)}{p_{1} (n) + \frac{q_{1} (n) q_{2} (n)}{p_{2} (n) + \frac{q_{2} (n) q_{3} (n)}{p_{3} (n) + \frac{q_{3} (n) q_{4} (n)}{p_{4} (n) + \frac{q_{4} (n) q_{5} (n)}{p_{5} (n) + \frac{q_{5} (n) q_{6} (n)}{p_{6} (n) + \frac{q_{6} (n) q_{7} (n)}{p_{7} (n) + \frac{}{}}}}}}}}}

= \frac{1}{q_{0} (n)} {a_{0} (n) + \frac{b_{1} (n)}{a_{1} (n) + \frac{b_{2} (n)}{a_{2} (n) + \frac{b_{3} (n)}{a_{3} (n) + \frac{b_{4} (n)}{a_{4} (n) + \frac{b_{5} (n)}{a_{5} (n) + \frac{b_{6} (n)}{a_{6} (n) + \frac{b_{7} (n)}{a_{7} (n) + \frac{}{}}}}}}}}},

giving convergents

{\frac{p_{0} (n + 1)}{q_{0} (n + 1)}, \frac{p_{1} (n + 1)}{q_{1} (n + 1)}, \frac{p_{2} (n + 1)}{q_{2} (n + 1)}, \frac{p_{3} (n + 1)}{q_{3} (n + 1)}, \frac{p_{4} (n + 1)}{q_{4} (n + 1)}, \frac{p_{5} (n + 1)}{q_{5} (n + 1)}, \frac{p_{6} (n + 1)}{q_{6} (n + 1)}, \frac{p_{7} (n + 1)}{q_{7} (n + 1)}, . . .}

x_{\infty} \equiv \lim_{n \to \infty} x_{n}

Weisstein, Eric W., Khinchin's Constant, from MathWorld—A Wolfram Web Resource.
Weisstein, Eric W., Continued Fraction Constants, from MathWorld—A Wolfram Web Resource.