Every now and then I stumble across a rather elementary question: $$\text{Let} \; (a_n)_{n \in \mathbb{N}} \; \text{be a sequence with} \lim_{n \to \infty} a_n = 0. \; \text{Is there a test that reliable tests for convergence of} \; \sum_\mathbb{N} a_n ?$$
I suspect the answer is no, since otherwise, such a test would be taught besides the root / ratio test in Calculus 1 & 2. However, each iteration of those courses I've witnessed and the books I consulted all tacitly leave this issue out. They rather state that the taught tests are simply unconclusive if the result is 1.
In the light of the plethory of different, more fine-tuned tests that exist, I'm happy with the following variation: $$ \text{Let} \; \alpha: n \mapsto a_n \; \text{be computable. Are there countably many computable tests} \; (T_n)_{n \in \mathbb{N}} \; \text{and a computable function} \; D \; \text{such that} \; T_{D(\{\alpha\})} \; \text{decides whether} \; \sum_\mathbb{N} a_n \; \text{converges?} $$
Here, I'm using $\{\alpha\}$ to denote a code for the function $\alpha$. I'm most interested in the case that the sequences are real, i.e., $a_n \in \mathbb{R}$, although that slightly complicates the computability part. If it's easier to deal with rational or algebraic sequences, I'll take the restriction as well, and if it gets easier with c.e. functions instead of computable ones, that's fine too.
