A new definition of the Adomian polynomials.

The article presents a study which identifies new proof of convergence for Adomian polynomials. The study uses Cauchy-Kovalevskaya theorem, Taylor expansion series and Banach-space analog to determine Adomian decompositions series for nonlinear ordinary and partial differential equations. Results show that four classes of Adomian polynomials belong to the common family of decompositions series that are uniformly convergent.