Definition:Convergent of Continued Fraction

Definition
Let $F$ be a field, such as the field of real numbers.

Let $n \in \N \cup \{\infty\}$ be an extended natural number.

Let $C = \left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$.

Let $k \leq n$ be a natural number.

Also see

 * Equivalence of Definitions of Convergent of Continued Fraction
 * Irrational Number is Limit of Unique Simple Infinite Continued Fraction, where it is shown that it is possible to talk directly about the convergents to any irrational number $x$.