It is not difficult to show that, given a computable sequence $(a_n)_{n=0}^\infty,$ one cannot (in general) compute if $\sum_{n=0}^\infty a_n$ converges. For example, given a program $\varphi,$ we can define the sequence $$(a_\varphi)_n=\begin{cases} 0 & \varphi\text{ halts in }n\text{ steps,}\\ 1 & \text{otherwise.}\end{cases}$$ In this case, $\sum_{n=0}^\infty(a_\varphi)_n$ converges if and only if $\varphi$ halts after some finite number of steps, so testing convergence is at least as hard the halting problem.
However, this example feels somewhat contrived because I am unlikely to run into sequences like $(a_\varphi)_n$ in the wild. So,
Is there a less contrived family of sequences?
For the sake of concreteness, let's say we allow the family of sequences generated by $\mathbb Z[x]$, $\exp$, $|\bullet|$, and closed under addition, subtraction, multiplication, and composition. If it helps, we can include $\pi$ and $i$ (e.g., to get $\sin(\pi n)$).