I am currently studying the book,
Elementary Analysis: The theory of Calculus (Kenneth A.Ross)
In the section of Taylor's Theorem, it states:
The proof was given directly after:
This proof is direct and natural, as I can kinda understand how the proof is thought up.
However, the book stated that it isn't the usual proof,
I understand each step of the step, but when I then try to get a general sense of the proof, I was confused. The proof sounds weird, at least to me. Of course it get me to the result, but I don't know how the author of the proof ever thought up the idea.
Below are what I have tried:
In the proof, $$F(t)=f(x)-R_n(f,t,x)+\frac{M(x-t)^n}{n!}\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space(1)$$ ,while $R_n(f,t,x)$ is the $n$-th remainder of taylor series expansion of $f$ about centre $t$, evaluate at $x$, so by $(1)$, $$F(t)=\sum_{k=0}^n \frac{f^{(k)}(t)}{k!}(x-t)^k+\frac{(x-t)^n}{n!}[M-f^{(n)}(t)]$$
So what I think the insight is:
We want a function $F$ on $(a,b)$ such that $F(0)=F(x)$, If $x=0$, $$F(0)=f(x)=f(x)-R_n(f,0,x)+\frac{M(x-0)^n}{n!}$$ So we observe that we may make $F$ in terms of the centre of Taylor series, i.e. variable $t$. So we (guess?) the definition of $F$.
I does not safisfy the explanation, can anyone give an explanation of the deep insight of the proof of the Taylor's Theorem?
Any help will be appreciate. Thank you!