Definition:Convergent of Continued Fraction

Definition
Let $C = \left[{a_1, a_2, a_3, \ldots, a_n}\right]$ or $\left[{a_1, a_2, a_3, \ldots}\right]$ be a continued fraction, either finite or infinite.

Then the $k$th convergent $C_k$ of $C$ is the value of the finite continued fraction:
 * $C_k = \left[{a_1, a_2, \ldots, a_k}\right]$

In the finite case it is of course taken as read that $k \le n$.

Also see

 * Definition:Value of Continued Fraction
 * Simple Infinite Continued Fraction Converges, where it is shown that the sequence of convergents of a SICF does indeed converge to a limit.


 * Uniqueness of Simple Infinite Continued Fraction, where it is shown that it is possible to talk directly about the convergents to any irrational number $x$.