On a New Two Point Taylor Expansion With Applications

Abstract

A new two point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two point Taylor expansion is applied to expressing two different functions, one of which has a finite interval of convergence and the other infinite interval of convergence. The conditions for convergence are derived and the results are compared with the results of single point Taylor expansions as well as two point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Results of the new expansion are compared with the single point Taylor expansions as well as the classical two point Taylor expansion. Generally speaking two point Taylor expansions better represent the real function when the series is truncated. The new two point expansion and the classical two point expansion produced identical results for the problems treated. An application of the series to solution of a variable coefficient differential equation is also treated.