The Remainder in Taylor's Formula

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.