利用微分中值定理“中间点”的渐近性改进Taylor公式

详细信息

Amelionation of Taylor's Formula by Using the Asymptotic Behavior of the Intermediate Point of the Mean-Value Theorem

摘要: 针对利用中值定理“中间点”的渐近性能改善Taylor公式,改善后的公式中间点是否还有渐近性进行了讨论.由讨论这个问题出发,证明了用这种近似式产生的误差,优于用同阶Taylor多项式产生的误差,并给出这个误差的一个类似于Taylor公式余项的表达式.然后研究了这种误差中的“中间点”的渐近性,给出了“中间点渐近性”的递归性的证明.本文并将讨论的结果推广到了多元Taylor公式.
- 渐近性 /
- 中间点 /
- ⊗ /
- 积
Abstract: May one obtain a more better approximate formula about f(x) if onsubstitutes the approximate valu (x)/((n+1)) for intermediate point ξ? The problem isdiscussed and the error caused by using the approximate formula proves to be superior to that caused by using the homogeneous Taylor's polynomial and an expression is proved to be similatr to the remainder in Taylor's formula. Also asymptotic behavior of intermediate point of the error is studied and proved and, and the result is extended to Taylor's formula of a function of several variabls.
- asymptotic behavior /
- intermediate point /
- ⊗ /
- product