Expii

# Extension: Lagrange 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 Lagrange mean-value form, is $R_n(x) = \frac{f^{(n+1)}(\xi_L)}{(n+1)!} (x-a)^{n+1}$ for some real number $$\xi_L$$ 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 $$x$$.).