Let $f(z)=\sum_{k=0}^{\infty}a_k z^k$ be a formal [power series](/page/Power%20Series) and let $m,n\ge 0$. There exist polynomials $p,q\in\mathbb C[z]$, not both zero, with $\deg p\le m$, $\deg q\le n$, and $q(z)f(z)-p(z)=O(z^{m+n+1})$ as $z\to 0$. A Padé approximant in the normalised sense $q(0)=1$ exists if and only if this finite homogeneous system has a solution with $q(0)\neq 0$.