New Expansion and Infinite Series

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