Chapter 3.1, Problem 22ES

Prove Taylor’s Theorem 1.14 by following the procedure in the proof of Theorem 3.3. [Hint: Let

$g\text{(}t\text{) = }f(t)-P(t)-[f(x)-P(x)].\frac{{(t-x_0)}^{n+1}}{{(x-x_0)}_{n+1}},$

where P is the nth Taylor polynomial, and use the Generalized Rolle’s Theorem 1.10.]

Theorem 1.14 (Taylor's Theorem)

Suppose f ∈ Cⁿ[a, b], f^{(n + 1)} exists on [a, b], and x₀ ∈ [a, b], For every x ∈ [a, b], there exists a number ξ(x) between x₀ and x with

f(x) = P_n(x) + R_n(x).

where

$\begin{array}{l}{P}_n(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{″}(x_0)}{2!}{(}^x+\cdots \frac{f^{(n)}(x_0)}{n!}{(}^x\\ \text{ = }{\displaystyle \underset{k=0}{\overset{n}{\sum }}\frac{f^{(k)}(x_0)}{k!}}{(}^x\end{array}$