A computable convergent sequence of rationals can have a non-computable rate of convergence. By a rate of convergence of a sequence $(q_k)_k$, I mean a function $f : \omega \rightarrow \omega$ such that for all $x\in\omega$ we have $m \ge f(x) \Rightarrow |q_m - q_{f(x)}|<2^{-x}$. This $f$ is $\Delta_2$ if $(q_k)_k$ is computable.
For example, if $q_k = 0.(K[k]) := \sum_{x \in K[k]}2^{-x}$, where $K[k]$ is $k$-th approximation of the halting problem, the minimum rate of convergence is a modulus of $K$ and Turing equivalent to $K$.
While I have some intuitive idea about monotonous sequences as above, I have difficulty understanding rate of convergence for non-monotonous sequences. In particular: can I control the rate of convergence of a computable non-monotonous sequence to be an arbitrary $\Delta_2$ function?