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：我们通过解决适当的初始价值问题来确定泰勒多边近似中的拉格兰格函数。 因此,我们确定剩余期限,然后用自然立方样来估计,这导致泰勒近似的质量显著提高。 我们观察到许多数量级近似的准确性有所提高,包括独立变量 x 位于相关趋同半径以外的情况。