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Question

The first thing you have to realize about proving Taylor's theorem is that there are infinitely many versions of Taylor's theorem: one for each possible expression of the remainder term. In other words, what a Taylor's theorem really is is a proof that a certain expression involving n gives the nth remainder term, i.e. the diffierence between the nth Taylor polynomial and the function it approximates.
With that in mind we can motivate the Lagrange remainder form by (A) setting out to simply find some expression for the remainder, (B) generalizing our first attempt and discovering an infinite class of expressions for the remainder, and finally (C)
settling on the Lagrange form since it is somehow the most beautiful and compelling form in this class (or, on more pragmatic grounds, since it promises to be the most useful).

Answer 1

The first thing you have to realize about proving Taylor's theorem is that there are infinitely many versions of Taylor's theorem: one for each possible expression of the remainder term. In other words, what a Taylor's theorem really is a proof that a certain expression involving n gives the nth remainder term, i.e. the difference between the nth Taylor polynomial and the function it approximates. 要证明泰勒定理你必须意识到的第一件事就是泰勒定理有无限多版本即一对每个可能表达的余项。换言之，真正的泰勒定理就是证明某个相关表述n给出第n个余项，也就是证明第n个泰勒多项式和该多项式所接近的功能之间的不同。

With that in mind we can motivate the Lagrange remainder form by (A) setting out to simply find some expression for the remainder, (B) generalizing our first attempt and discovering an infinite class of expressions for the remainder, and finally (C)
settling on the Lagrange form since it is somehow the most beautiful and compelling form in this class (or, on more pragmatic grounds, since it promises to be the most useful).
知道了这个定理，我们可以激发拉格朗日余项格式通过(A)着手简单地找出余项的某种表述；（B）归纳我们的初步尝试发现一个无限级的余项表达；最后（C）决定拉各朗日格式。毕竟拉各朗日格式是那种在本极限里最美观和最有说服力的格式（或者，从更现实的基础来说拉各朗日格式是最实用的）。

Answer 2

首先，你必须认识到约证明泰勒定理的是，有无穷多个版本的泰勒定理：每个可能表达的剩余任期。换句话说，什么是泰勒定理真的是一个证据，证明某个表达式涉及ñ使第n个剩余任期，即diffierence之间的n次方泰勒多项式和它的功能接近。

考虑到这一点我们可以激发拉格朗日其余形式包括： （一）制定了简单地找到了一些表达的剩余时间， （乙）推广我们的第一次尝试和探索的无限级的表现形式的剩余，并最终（丙）

解决问题的Lagrange形式，因为它是最美丽的多少和有说服力的形式在本级（或上更加务实的理由，因为它有望成为最有用的） 。

Answer 3

您必须体会关于证明泰勒的定理的第一件事是无限地有泰勒的定理的许多版本： 一余项的每个可能的表示的。 真正换句话说，泰勒的定理is is介入n的某一表示给第n余项的证明，即在多项第n的泰勒和作用之间的diffierence它接近。
我们可以通过(a)发现剩下的人的某一表示的开始鉴于此刺激拉格朗日剩下的人形式， (b)推断我们的第一个尝试和发现表示无限类剩下的人的和最后(c) 在拉格朗日形式的settling，因为它莫名其妙地是在这类的最美好和最强制的形式(或，在更加重实效的地面，因为它许诺是最有用的)。