**finite continued fraction**(FCF) is a map $$f: \mathbf{N}^m \rightarrow \mathbf{Q} $$ $$\left( a_1, a_2, \cdots a_m \right) \mapsto a_1 + \frac{1}{a_2 + \frac{1}{\ddots + \frac{1}{a_m}}}$$

Continued fractions are calculated by creating a table of convergents, as follows:

$k$ | $a_k$ | $p_k$ | $q_k$ | $C_k$ |

$-1$ | $0$ | |||

$0$ | $1$ | $0$ | ||

$1$ | $a_1$ | $a_1 \cdot p_0 + p_{-1}$ | $1$ | $\frac{p_1}{q_1}$ |

$k$ | $a_k$ | $a_k \cdot p_{k-1} + p_{k-2}$ | $a_k \cdot q_{k-1} + q_{k-2}$ | $\frac{p_k}{q_k}$ |

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also: Continued fractions (1)

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