I thought of this question in connection with Calculus II (a course in the US which includes, among other things, techniques of integration and convergence tests for series, both of which are taught as a bunch of techniques that might work for a particular problem) but feel free to make the answers as complicated as necessary. The question of whether an elementary antiderivative of an elementary function $f(x)$ exists is answered by the Risch algorithm and I want to ask a similar question about convergence of series.
Let $f(x)$ be an elementary function and $a_n=f(n)$ for $n=0,1,2,\ldots$. Is there an algorithm that will tell us if $\sum_{n=0}^\infty a_n$ converges? Are there examples that are undecidable?
The integral test might help convert a solution in one problem to the other but I don't see that it solves the problem entirely.