关于Taylor定理的余项

The Remainder in Taylor's Formula

讨论了利用微分中值定理“中间点”的渐近性，改进Taylor公式的问题；证明了在f（x0＋h）的Taylor公式中，用中间点ξ的渐近值x0+Ah替换余项中的ξ，可以得到一个推广的Taylor公式，而且推广后公式的余项中的中间点η也有渐近性．同时，证明了“中间点”渐近性的递推性，给出了一个更一般的结果（定理二）．

The way to utilize the progressive feature of the intermediate point of differential mean-value theorem to improve Taylor's formula is discussed. It is proved that an extended Taylor's formula can be obtained by using the progressive value of the intermediate point ξ, x0 + Ah, as a substitute for ξ in the remainder in Taylor's formula of f(x0 + h ). And it is also proved that the intermediate point η in the remainder of the extended formula is of progression. Besides the recursion feature of the intermediate point progression is proved, and a more general result is given.