Continued fraction convergents

Continued fraction convergents
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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 ${\frac {{\sqrt {a}}+m}{n}}$, specified by three positive integers $a,m,n$, where $a$ is not a perfect square, output the first $k$ convergents when given a positive number $k$.