Consider a periodic continued fraction $$x=\cfrac{a_1}{b_1+ \cfrac{a_2}{b_2+ \cfrac{\ddots}{\ddots b_{k-1}+ \cfrac{a_1}{b_1 + \ddots }}} }$$ for some sequence $a_1,\cdots, a_k$ and $b_1,\cdots, b_{k-1}$.
The wikipedia page on the "convergence problem":
https://en.wikipedia.org/wiki/Convergence_problem#Periodic_continued_fractions
gives an if and only if condition for $x$ to converge in terms of the roots of a quadratic involving the $k$th and $(k-1)$th convergents of $x$. Unfortunately, the reference is 120 years old and written in German.
We are looking for a modern reference, or a short proof of this fact.