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.

翻译：我们通过求解适当的初值问题，确定了泰勒多项式逼近中的拉格朗日函数。由此得到余项，并进一步采用自然三次样条进行逼近。该方法显著提升了泰勒逼近的质量。我们观察到逼近精度提高多个数量级，甚至在自变量x超出相应收敛半径的情况下也是如此。