Continued fraction convergents

Continued fraction convergents is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

This task has been flagged for clarification. Code on this page in its current state may be flagged incorrect once this task has been clarified. See this page's Talk page for discussion.

Given a positive real number, if we truncate its continued fraction representation at a certain depth, we obtain a rational approximation to the real number. The sequence of successively better such approximations is its convergent sequence.

Problem:

• Given a positive rational number ${\frac {m}{n}}$, specified by two positive integers $m,n$, output its entire sequence of convergents.
• Given a quadratic real number ${\frac {b{\sqrt {a}}+m}{n}}>0$, specified by integers $a,b,m,n$, where $a$ is not a perfect square, output the first $k$ convergents when given a positive number $k$.

The output format can be whatever is necessary to represent rational numbers, but it probably should be a 2-tuple of integers.

For example, given $a=2,b=1,m=0,n=1$, since

$${\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}$$

the program should output $(1,1),(3,2),(7,5),(17/12),(41/29),\dots$.

A simple check is to do this for the golden ratio ${\frac {{\sqrt {5}}-1}{2}}$, that is, $a=5,b=1,m=-1,n=2$, which would just output the Fibonacci sequence.