一种用定积分证明泰勒中值定理的方法

A Proof of Taylor's Theorem Using Definite Integral

介绍一种从拉格朗日中值定理出发,通过重复的定积分运算,直接推导出泰勒中值定理的方法.首先通过定义函数,将拉格朗日中值定理的表达式转化为可积的形式,之后在定义域的任一闭区间上对其进行定积分;得到的结果通过定义第二个函数再次转化为可积的形式,继续进行定积分.如此循环n-1次,得到一个类似于泰勒中值定理的n次多项式.通过对在定积分过程当中所定义的n个函数的值域进行讨论,便可将此n次多项式转化为泰勒中值定理的形式.这种方法不需要泰勒公式作为推导的基础,因此,它能较好地揭示泰勒中值定理的本质,建立泰勒中值定理与拉格朗日中值定理之间的有机关联.

With Lagrange＇s theorem been treated as the starting point,a method includes repeated definite integral can work out Taylor＇s theorem directly. Transform the expression of Lagrange＇s theorem to make it integrable using the function defined in the first place. Then do the integral on any closed interval of the domain of definition; transform the result to make it integrable by defining the second function,then do the integral again. Cycle for n-1 times like this,a n-order polynomial which is similar to the expression of Taylor＇s theorem will be gained. By discussing the range of these n functions, the polynomial can be transformed into the form of Taylor＇s theorem. The Taylor＇s formula is not the foundation of this method, so this method can reveal the essence of Taylor＇s theorem,can build an organic relationship between Lagrange＇s theorem and Taylor＇s theorem.