Extension: Cauchy Mean Value Form of the Error in Taylor Approximation - Expii

If \(T_n(x)\) is the degree \(n\) Taylor approximation of \(f(x)\) at \(x=a\), then the approximation error \(R_n(x) = f(x) - T_n(x)\) (sometimes called remainder), in Cauchy mean-value form, is \[ R_n(x) = \frac{f^{(n+1)}(\xi_C)}{n!} (x-\xi_C)^n (x-a)\] for some real number \(\xi_C\) between \(a\) and \(x\). (Technical hypotheses: this formula holds if \(f^{(n)}\) is continuous on the closed interval between \(a\) and \(x\), and differentiable on the open interval between \(a\) and \(x\).).