Under what conditions, the function \(f(x) \) is equal to the Taylor series generated by \(f\) at \(x = a\text{?}\) That is, when can we say \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n\text{?}\)
If the Taylor polynomial of degree \(n\text{,}\)\(p_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k\) , is used as an approximation to \(f(x)\text{,}\) how can we estimate the error in this approximation? That is, how can we estimate the magnitude of \(R_n(x) = f(x) - p_n(x)\text{?}\)
Theorem 2.1 and Theorem 2.5 will enable us to answer these questions. We will begin by stating Taylor’s formula and the remainder estimation theorem, and then we will discuss some applications of these theorems. The proof of the remainder theorem is postponed to the end of this section because it requires some technical details that are not necessary for understanding the main ideas of this section.
