Description of Taylor's Theorem and Applications
Taylor's Theorem Let $k \ge 1$ be an integer and let the function $f: \mathbf{R} \rightarrow \mathbf{R}$ be $k$ times differentiable at the point $a \in \mathbf{R}$. Then there exists a function $h_k: \mathbf{R} \rightarrow \mathbf{R}$ such that $$ f(x) = f(a) + f'(a)(x-a) + \frac{f^{''}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k + h_k(x) (x-a)^k, $$ and $$ \lim_{x\rightarrow a} h_k(x) = 0 $$
Mean-value forms of the reminder Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be $k+1$ times differentiable on the open interval with $f^{(k)}$ continuous on the closed interval between $a$ and $x$. Then $$ R_k(x) = \frac{f^{(k+1)} (c)}{(k+1)!} (x-a)^{k+1} $$ for some $c$ between $a$ and $x$. Moreover, if there exists a real number $M$ such that $f^{(k+1)} \le M$ for all $x$, then $$ R_k(x) \le \frac{M}{(k+1)!} \vert x-a\vert^{k+1} $$
Proof Le $$g(t)$$ be the difference between $$(f - p) - R_n$$ where $$p$$ is $$k$$-th-order Taylor polynomial of $$f(x)$$ at $$x=t$$. $$ g(t) = f(x) - (f(t) + f'(t)(x-t) + \frac{f^{(2)}(t)}{2!}(x-t)^2 + \cdots + \frac{f^{(k)}(t)}{k!}(x-t)^k - R_k(x) \frac{(x-t)^{k+1}}{(x-a)^{k+1}} $$ We have 1. $$g(x) = f(x) - f(x) + 0 +\cdots + 0 = 0 $$ 2. $$g(a) = f(x) - (f(a) + f'(t)(x-a) + \frac{f^{(2)}(t)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(t)}{k!}(x-a)^k - R_k(x) \frac{(x-a)^{k+1}}{(x-a)^{k+1}} $$. Then $g(a) = f(x) - p_k(x) - R_k(x) = 0$ As $g(t)$ satisfies Rolle's theorem, there exists $c$ between $a$ and $x$ such that $g'(c)=0$. With some derivations, we have $$ R_k(x) = \frac{f^{k+1}(c)}{(k+1)!} (x-a)^{k+1} $$.