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# Convergents constant

This (probably original) research topic requires further investigation.

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

## Iterated continued fractions from convergents

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

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

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

${\displaystyle x_{0}=[a_{0}(0);a_{1}(0),a_{2}(0),\dots ]=a_{0}(0)+{\cfrac {1}{a_{1}(0)+{\cfrac {1}{a_{2}(0)+{\cfrac {1}{a_{3}(0)+{\cfrac {1}{a_{4}(0)+{\cfrac {1}{a_{5}(0)+{\cfrac {1}{a_{6}(0)+{\cfrac {1}{a_{7}(0)+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}},\,}$

giving convergents

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

then

${\displaystyle x_{1}={\tfrac {p_{0}(1)}{q_{0}(1)}}+{\cfrac {1}{{\tfrac {p_{1}(1)}{q_{1}(1)}}+{\cfrac {1}{{\tfrac {p_{2}(1)}{q_{2}(1)}}+{\cfrac {1}{{\tfrac {p_{3}(1)}{q_{3}(1)}}+{\cfrac {1}{{\tfrac {p_{4}(1)}{q_{4}(1)}}+{\cfrac {1}{{\tfrac {p_{5}(1)}{q_{5}(1)}}+{\cfrac {1}{{\tfrac {p_{6}(1)}{q_{6}(1)}}+{\cfrac {1}{{\tfrac {p_{7}(1)}{q_{7}(1)}}+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,}$
${\displaystyle ={\frac {1}{q_{0}(1)}}\left\{p_{0}(1)+{\cfrac {q_{0}(1)q_{1}(1)}{p_{1}(1)+{\cfrac {q_{1}(1)q_{2}(1)}{p_{2}(1)+{\cfrac {q_{2}(1)q_{3}(1)}{p_{3}(1)+{\cfrac {q_{3}(1)q_{4}(1)}{p_{4}(1)+{\cfrac {q_{4}(1)q_{5}(1)}{p_{5}(1)+{\cfrac {q_{5}(1)q_{6}(1)}{p_{6}(1)+{\cfrac {q_{6}(1)q_{7}(1)}{p_{7}(1)+{\cfrac {q_{7}(1)q_{8}(1)}{\ddots }}}}}}}}}}}}}}}}\right\}\,}$

${\displaystyle ={\frac {1}{q_{0}(1)}}\left\{a_{0}(1)+{\cfrac {b_{1}(1)}{a_{1}(1)+{\cfrac {b_{2}(1)}{a_{2}(1)+{\cfrac {b_{3}(1)}{a_{3}(1)+{\cfrac {b_{4}(1)}{a_{4}(1)+{\cfrac {b_{5}(1)}{a_{5}(1)+{\cfrac {b_{6}(1)}{a_{6}(1)+{\cfrac {b_{7}(1)}{a_{7}(1)+{\cfrac {b_{8}(1)}{\ddots }}}}}}}}}}}}}}}}\right\},\,}$

giving convergents

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

${\displaystyle \cdots \,}$

${\displaystyle x_{n}={\tfrac {p_{0}(n)}{q_{0}(n)}}+{\cfrac {1}{{\tfrac {p_{1}(n)}{q_{1}(n)}}+{\cfrac {1}{{\tfrac {p_{2}(n)}{q_{2}(n)}}+{\cfrac {1}{{\tfrac {p_{3}(n)}{q_{3}(n)}}+{\cfrac {1}{{\tfrac {p_{4}(n)}{q_{4}(n)}}+{\cfrac {1}{{\tfrac {p_{5}(n)}{q_{5}(n)}}+{\cfrac {1}{{\tfrac {p_{6}(n)}{q_{6}(n)}}+{\cfrac {1}{{\tfrac {p_{7}(n)}{q_{7}(n)}}+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,}$
${\displaystyle ={\frac {1}{q_{0}(n)}}\left\{p_{0}(n)+{\cfrac {q_{0}(n)q_{1}(n)}{p_{1}(n)+{\cfrac {q_{1}(n)q_{2}(n)}{p_{2}(n)+{\cfrac {q_{2}(n)q_{3}(n)}{p_{3}(n)+{\cfrac {q_{3}(n)q_{4}(n)}{p_{4}(n)+{\cfrac {q_{4}(n)q_{5}(n)}{p_{5}(n)+{\cfrac {q_{5}(n)q_{6}(n)}{p_{6}(n)+{\cfrac {q_{6}(n)q_{7}(n)}{p_{7}(n)+{\cfrac {q_{7}(n)q_{8}(n)}{\ddots }}}}}}}}}}}}}}}}\right\}\,}$

${\displaystyle ={\frac {1}{q_{0}(n)}}\left\{a_{0}(n)+{\cfrac {b_{1}(n)}{a_{1}(n)+{\cfrac {b_{2}(n)}{a_{2}(n)+{\cfrac {b_{3}(n)}{a_{3}(n)+{\cfrac {b_{4}(n)}{a_{4}(n)+{\cfrac {b_{5}(n)}{a_{5}(n)+{\cfrac {b_{6}(n)}{a_{6}(n)+{\cfrac {b_{7}(n)}{a_{7}(n)+{\cfrac {b_{8}(n)}{\ddots }}}}}}}}}}}}}}}}\right\},\,}$

giving convergents

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

### Convergence of iterated continued fractions from convergents

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

${\displaystyle x_{\infty }\equiv \lim _{n\to \infty }x_{n}\,}$