Homotopy Analysis Method itsApplications in Mathematics

In this paper, the basic ideas of a new nonlinear analytical technique, namely Homotopy Analysis Method, are described, and a new family of real functions, called approaching functions, is introduced. Some basic properties of this family of real functions are given. It is rigorously proved that the convergence region of the traditional Taylor series can be greatly enlarged by multiplying its each term by ap- proaching functions of this family one after another in order.