Is there a general way to find the sum
$\sum_{n=1}^{\infty}\frac1{P(n)}$
where $P(x)$ is a polynomial of degree $k\geq 2$, with coefficients $a_0,a_1,\dots,a_k$? (possibly restricted to integers)
What is $\sum\limits_{n=1}^{\infty}\frac1{2n^7+n^3-5}$?
And how to find $\sum\limits_{n=1}^{\infty}\frac{Q(n)}{P(n)}$ for two polynomials?