Exact approximation order of real numbers in Cantor series expansions

Let $Q=\{q_n\}_{n\geq1}$ be a sequence of integers with $q_n\geq2$ for every $n\geq1$. Every $x\in[0,1)$ admits a Cantor series expansion of the form $$ x = \frac{\varepsilon_1(x)}{q_1} + \frac{\varepsilon_2(x)}{q_1q_2} +\cdots+ \frac{\varepsilon_n(x)}{q_1q_2\cdots q_n} +\cdots.$$In th