Enhancing the Accuracy of the Taylor Polynomial by determining the Remainder Term

We determine the Lagrange function in Taylor polynomial approximation by solving an appropriate initial-value problem. Hence, we determine the remainder term which we then approximate by means of a natural cubic spline. This results in a significant improvement in the quality of the Taylor approximation. We observe improvements in the accuracy of the approximation of many orders of magnitude, including a case when the independent variable x lies beyond the relevant radius of convergence.