In rings ${\Gamma }_M$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left(M_l\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma }_M$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to ${\Gamma }_M$, provided ${\Gamma }_M$ is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...