Power Expansions of Solutions of One Algebraic or Differential Equation

We give the generalizations of the classical theorems of Cauchy and Cauchy-Kovalevskaya on the existence and uniqueness of the analytical solution of one analytical or differential equation. The generalizations are given in terms of Power Geometry and describe situations when we can stop the computation of following terms of the local or asymptotic power expansion of a solution of one equation. We give also further generalizations show how to apply the theory in the complicated cases. As an example, we consider the first Painleve equation.