Let $r(n)=p(n)/q(n)$ be a rational function, where each polynomial has integer coefficients, and assume that $q$ has no integer roots for $n\geq 1$. Then $S=\sum_{n=1}^{\infty} r(n)$ converges if and only if $\deg(q)-\deg(p)\ge 2$. My question is: is there an easy way to know for which such $r(n)$ does a closed-form for $S$ exist? For example, I know that if $p(n)\equiv 1$ and $q(n)=n^{2k}$, one could use Euler's strategy for computing $\sum_{n=1}^{\infty} n^{-2k}$ to find a closed form. Likewise, I know there's a closed form for $\sum_{n=1}^{\infty} (n^2+1)^{-1}$ by complex-analysis, $\sum_{n=2}^{\infty} (n^2-1)^{-1}$ by telescoping, etc., but I was wondering if there's a more systematic approach.
It seems like the corresponding problem for integration is easier at least because of partial fraction decomposition. In that case, we can decompose and integrate as a finite sum of reciprocals, logs, and arctangents, but I have no clue if we can do the same thing with series.