If you're worried about convergence... You can rewrite your equation as $g(z) = {1 \over f(1/z)} = 1/n$ for $z$ near ${1 \over n}$. $g(z)$ is thus $z^d$ divided by a polynomial with a nonzero constant term. Here $d$ is the degree of $f(z)$. Hence $g(z) = h(z)^d$ for some $h(z)$ with $h(0) = 0$ and $h'(0) \neq 0$.  So you're solving $h(z) = n^{-{1 \over d}}$  which has a convergent power series solution in $n^{-{1 \over d}}$ by inverting $h(z)$.