Extensão: Teorema do valor médio de Cauchy

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 $\xi_C$ strictly 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 $x$.)