The Remainder in Taylor's Formula
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摘要: 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).Abstract: 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.
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Keywords:
- asymptotic behavior /
- intermediate point /
- meanvalue theorem
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