When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$? This is a follow up on a previous question of mine.
Out of curiosity, I am wondering more generally when a closed form exists for
$$\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$$
where $P$ and $Q$ are both polynomials. Obviously for convergence $\deg(Q)\ge \deg(P)+2$, but I don't know much from there.
 A: As GH from MO says, it depends what you think is a closed form.
Any function of the form $P(n)/Q(n)$ with $\deg P \leq \deg Q -2$ can be written as a linear combination of $1/(n+1) - 1/(n+\alpha)$ and $1/(n+\alpha)^k$, for various $\alpha$ and various $k>1$. (Use partial fraction decomposition. The bound on $\deg P$  ensures that the coefficients of $1/(n+1)$ will cancel out.)
The sum $\sum \left( \frac{1}{n+1} - \frac{1}{n+\alpha} \right)$ is $\gamma+\psi(\alpha)$, where $\psi$ is the digamma function. The sum $\sum \frac{1}{(n+\alpha)^k}$ is the polygamma function $\psi^k(z)$. In both cases, it is worth noting the recurrence identities and reflection relations in the linked Wikipedia articles, which let you relate values at $\alpha$ to values at $m \pm \alpha$, for $m$ an integer.
A: Closed form is a subjective term. At any rate, our understanding of these sums is quite limited, e.g. it is not known if $\sum_{n=1}^\infty n^{-5}$ is an irrational number (same for any smaller negative odd exponent).
