New expansion and infinite series

Abstract

Different from the Taylor polynomials, a new formula for function expansion is proposed where the terms are not polynomials. A new infinite series based on the new formula is also proposed, and the new infinite series can keep some important properties of the original functions. Some forms of remainder are also presented for analysis of convergence. In order to show some internal relationships between the new results and Taylor's theorem, some important theorems have been proved in this paper. Finally, Some examples are given and the regions of convergence for the new infinite series are analyzed, the results show that the region of convergence is much larger than that obtained by Taylor's Series.Mathematics Subject Classification: 41A58, 41A20, 41A30, 40A30

General Methods For Solving Ordinary Differential Equations 1

“…This section contains two subsections. Subsection 2.1 gives a new proof of the new mathematical expansion proposed in paper [1]. Compared with document [1], the new proof is simpler.…”

“…Subsection 2.1 gives a new proof of the new mathematical expansion proposed in paper [1]. Compared with document [1], the new proof is simpler. In subsection 2.2, based on my new mathematical expansion [1], two important theorems are proved, they are the foundations for constructing differential equations, which can be used for solving the kth order linear ordinary differential equations (see section3).…”

“…An important theorem concerning a new mathematical expansion for a given function f (x) has been proposed in my paper [1] and given in the following:…”

“…Based on the new mathematical expansion I proposed in [1] (Daiyuan Zhang, 2014), this paper studies a new method for solving the kth order linear ordinary differential equations, where k is a positive integer.…”

“…In this paper, I present a unified method for solving linear ordinary differential equations. The original foundation of this paper is the new mathematical expansion proposed in my paper [1], without this new mathematical expansion [1], we can not get the new method for solving the high order linear ordinary differential equations. This method is completely different from the known traditional methods.…”