My question is about rational functions of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x) = \sum_{i=0}^n p_i x^i$ and $q(x) = \sum_{i=0}^n q_i x^i$ with $p_i, q_i \in \mathbb{Q}$ and $q_0 \ne 0$. Consider the power series expansion (equivalently, Taylor expansion at $x = 0$) given by $f(x) = \sum_{i=0}^\infty a_i x^i$. Is there an algorithm that takes $p_0,\dots,p_n,q_0,\dots,q_n$ as input and decides whether $a_i \ge 0$ for all $i \in \mathbb{N}$?
I know how to solve this for the special case of $q(x) = 1 - c x$ for $c > 0$. Then $f(x) = \sum_{i=0}^\infty \sum_{j=0}^n p_j x^j c^i x^i = \sum_{k=0}^\infty \left(\sum_{j=0}^{\min(k,n)} p_j q_1^{-j} \right)c^kx^k$. So the coefficients are nonnegative iff $0 \le \sum_{j=0}^k p_j c^{-j}$ for all $1 \le k \le n$, which is easy to check.
What is the algorithm for the general case?
You probably noticed that the coefficients are a constant recursive sequence. The coefficients given by those of the denominator. Conversely, if you have a sequence $a_n$ that is a constant recursive sequence, then those are the coefficients of the series of a rational function. In your case all coefficients are rational.
See here for algorithms for low degrees ($\leq5$) of the denominator, but less hopeful show decidability for larger degree.