Let $f$ be a polynomial function of degree at least $2$ with integer coefficients, and assume that $f(n)$ is nonzero for any positive integer $n$.

**Question:** Is it algorithmically decidable whether
$$
S(f) \ := \ \sum_{n = 1}^\infty \frac{1}{f(n)}
$$
is rational or not? -- Which are the known necessary or sufficient criteria
for the rationality or the irrationality of the value of this expression?

*Examples:* $S(n^2) = \zeta(2) = \frac{\pi^2}{6}$ and $S(n^3) = \zeta(3)$
are irrational, while $S(n^2+n) = 1$ is rational.

notinclude $n^{2k+1}$. $\endgroup$ – Stefan Kohl Oct 2 '16 at 22:17provethat it is decidable. I think you're really just interested in which cases the irrationality isknown. $\endgroup$ – Timothy Chow Oct 3 '16 at 18:04