Finding elements of $K[[x]]\setminus K[x]_{(x)}$.

Question

Let $K$ be a field. We have that the elements of $K[x]_{(x)}$ are of the form $\frac{f}{g}$ for some $f,g\in K[x]$ such that $g(0)\neq 0$. Since the units of $K[[x]]$ are the elements not vanishing at $0$, we may understand $\frac{f}{g}$ as the element $fg^{-1}$ of $K[[x]]$. It follows that $K[x]_{(x)}\subset K[[x]]$. Is there an easy way to determine if $\sum_{i=0}^{\infty}a_i x^i\in K[[x]]$ belongs to $K[x]_{(x)}$ or not?