Abstract
The arithmetic operations and functions of Taylor series can be defined by C++ language. The functions which consist of arithmetic operations, pre-defined functions and conditional statements can be expanded in Taylor series. Using this, the solution of an ordinary differential equation can be expanded in Taylor series. The solution can be expanded up to arbitrary order, so the calculation formula of arbitrary order can be used instead of Runge-Kutta formula. Taylor series can be used for the evaluations of the errors and the optimal step size within given error allowance easily. In addition, we can transform Taylor series into Pade series, which give arbitrary order, high precision and A-stable formula for solving ordinary differential equation numerically.
