Convergents constant

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

Iterated continued fractions from convergents

Cf. Iterated continued fractions from convergents.

In order to get from iterate $\scriptstyle x_n \,$ to iterate $\scriptstyle x_{n+1} \,$

Express $\scriptstyle x_n \,$ with the convergents $\scriptstyle c_i(n-1) \,$ of $\scriptstyle x_{n-1} \,$ as a continued fraction $\scriptstyle [c_0(n);\, c_1(n),\, c_2(n),\, \ldots] \,=\, \frac{1}{q_0(n)}\,[p_0(n);\, q_0(n)\,q_1(n)\,/\,p_1(n),\, q_1(n)\,q_2(n)\,/\,p_2(n),\, ...]; \,$
Compute the convergents $\scriptstyle c_0(n+1) \,\equiv\, \frac{p_0(n+1)}{q_0(n+1)} \,=\, \frac{1}{q_0(n)} \, \left(p_0(n)\right),\, c_1(n+1) \,\equiv\, \frac{p_1(n+1)}{q_1(n+1)} \,=\, \frac{1}{q_0(n)} \left( \, p_0(n) + \frac{q_0(n)\,q_1(n)}{p_1(n)} \right),\, c_2(n+1) \,\equiv\, \frac{p_2(n+1)}{q_2(n+1)} \,=\, \frac{1}{q_0(n)} \left( \, p_0(n) + \frac{q_0(n)\,q_1(n)}{p_1(n) + \frac{q_1(n)\,q_2(n)}{p_2(n)}} \right),\, \ldots \,$
(using the efficient recursive method shown on generalized continued fractions convergents;)
The next iterate is $\scriptstyle x_{n+1} \,=\, [c_0(n+1);\, c_1(n+1),\, c_2(n+1),\, \ldots]. \,$

For example, starting with $\scriptstyle x_0 \,\equiv\, x \,$ , where $\scriptstyle x \,$ is a positive real constant, first obtain the simple continued fraction

$x_0 = [a_0(0);a_1(0),a_2(0),\dots]= a_0(0) + \cfrac{1}{a_1(0) + \cfrac{1}{a_2(0) + \cfrac{1}{a_3(0) + \cfrac{1}{a_4(0) + \cfrac{1}{a_5(0) + \cfrac{1}{a_6(0) + \cfrac{1}{a_7(0) + \cfrac{1}{\ddots}}}}}}}}, \,$

giving convergents

$\Bigg\{ \frac{p_0(1)}{q_0(1)}, \frac{p_1(1)}{q_1(1)}, \frac{p_2(1)}{q_2(1)}, \frac{p_3(1)}{q_3(1)}, \frac{p_4(1)}{q_4(1)}, \frac{p_5(1)}{q_5(1)}, \frac{p_6(1)}{q_6(1)}, \frac{p_7(1)}{q_7(1)}, ... \Bigg\} \,$

then

$x_1 = \tfrac{p_0(1)}{q_0(1)} + \cfrac{1}{\tfrac{p_1(1)}{q_1(1)} + \cfrac{1}{\tfrac{p_2(1)}{q_2(1)} + \cfrac{1}{\tfrac{p_3(1)}{q_3(1)} + \cfrac{1}{\tfrac{p_4(1)}{q_4(1)} + \cfrac{1}{\tfrac{p_5(1)}{q_5(1)} + \cfrac{1}{\tfrac{p_6(1)}{q_6(1)} + \cfrac{1}{\tfrac{p_7(1)}{q_7(1)} + \cfrac{1}{\ddots}}}}}}}} \,$

$= \frac{1}{q_0(1)} \left\{ p_0(1) + \cfrac{q_0(1) q_1(1)}{p_1(1) + \cfrac{q_1(1) q_2(1)}{p_2(1) + \cfrac{q_2(1) q_3(1)}{p_3(1) + \cfrac{q_3(1) q_4(1)}{p_4(1) + \cfrac{q_4(1) q_5(1)}{p_5(1) + \cfrac{q_5(1) q_6(1)}{p_6(1) + \cfrac{q_6(1) q_7(1)}{p_7(1) + \cfrac{q_7(1) q_8(1)}{\ddots}}}}}}}} \right\} \,$

$= \frac{1}{q_0(1)} \left\{ a_0(1) + \cfrac{b_1(1)}{a_1(1) + \cfrac{b_2(1)}{a_2(1) + \cfrac{b_3(1)}{a_3(1) + \cfrac{b_4(1)}{a_4(1) + \cfrac{b_5(1)}{a_5(1) + \cfrac{b_6(1)}{a_6(1) + \cfrac{b_7(1)}{a_7(1) + \cfrac{b_8(1)}{\ddots}}}}}}}} \right\}, \,$

giving convergents

$\Bigg\{ \frac{p_0(2)}{q_0(2)}, \frac{p_1(2)}{q_1(2)}, \frac{p_2(2)}{q_2(2)}, \frac{p_3(2)}{q_3(2)}, \frac{p_4(2)}{q_4(2)}, \frac{p_5(2)}{q_5(2)}, \frac{p_6(2)}{q_6(2)}, \frac{p_7(2)}{q_7(2)}, ... \Bigg\} \,$

$\cdots \,$

$x_n = \tfrac{p_0(n)}{q_0(n)} + \cfrac{1}{\tfrac{p_1(n)}{q_1(n)} + \cfrac{1}{\tfrac{p_2(n)}{q_2(n)} + \cfrac{1}{\tfrac{p_3(n)}{q_3(n)} + \cfrac{1}{\tfrac{p_4(n)}{q_4(n)} + \cfrac{1}{\tfrac{p_5(n)}{q_5(n)} + \cfrac{1}{\tfrac{p_6(n)}{q_6(n)} + \cfrac{1}{\tfrac{p_7(n)}{q_7(n)} + \cfrac{1}{\ddots}}}}}}}} \,$

$= \frac{1}{q_0(n)} \left\{ p_0(n) + \cfrac{q_0(n) q_1(n)}{p_1(n) + \cfrac{q_1(n) q_2(n)}{p_2(n) + \cfrac{q_2(n) q_3(n)}{p_3(n) + \cfrac{q_3(n) q_4(n)}{p_4(n) + \cfrac{q_4(n) q_5(n)}{p_5(n) + \cfrac{q_5(n) q_6(n)}{p_6(n) + \cfrac{q_6(n) q_7(n)}{p_7(n) + \cfrac{q_7(n) q_8(n)}{\ddots}}}}}}}} \right\} \,$

$= \frac{1}{q_0(n)} \left\{ a_0(n) + \cfrac{b_1(n)}{a_1(n) + \cfrac{b_2(n)}{a_2(n) + \cfrac{b_3(n)}{a_3(n) + \cfrac{b_4(n)}{a_4(n) + \cfrac{b_5(n)}{a_5(n) + \cfrac{b_6(n)}{a_6(n) + \cfrac{b_7(n)}{a_7(n) + \cfrac{b_8(n)}{\ddots}}}}}}}} \right\}, \,$

giving convergents

$\Bigg\{ \frac{p_0(n+1)}{q_0(n+1)}, \frac{p_1(n+1)}{q_1(n+1)}, \frac{p_2(n+1)}{q_2(n+1)}, \frac{p_3(n+1)}{q_3(n+1)}, \frac{p_4(n+1)}{q_4(n+1)}, \frac{p_5(n+1)}{q_5(n+1)}, \frac{p_6(n+1)}{q_6(n+1)}, \frac{p_7(n+1)}{q_7(n+1)}, ... \Bigg\} \,$

Convergence of iterated continued fractions from convergents

We define the limit of iterated continued fractions from convergents for a constant $\scriptstyle x \,$ as

$x_{\infty} \equiv \lim_{n \to \infty} x_n \,$