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 \(R_n(x) = f(x) - p_n(x)\text{?}\)
