Definition:Convergent of Continued Fraction

Definition
Let $F$ be a field.

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.

Then 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

 * Definition:Value of Continued Fraction
 * Simple Infinite Continued Fraction Converges to Irrational Number
 * 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 $x$.