Definition:Convergent of Continued Fraction/Definition 1

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.

The $k$th convergent $C_k$ of $C$ is the value of the finite continued fraction:
 * $C_k = \left[{a_0, a_1, \ldots, a_k}\right]$

Also see

 * Equivalence of Definitions of Convergent of Continued Fraction