Lagrange Error Bound - Expii

If \(T_n(x)\) is the degree \(n\) Taylor approximation of \(f(x)\) at \(x=a\), then the Lagrange error bound provides an upper bound for the error \(R_n(x) = f(x) - T_n(x)\) for \(x\) close to \(a\). Namely, if \(|f^{(n+1)}(x)| \le M\) on an interval \(I = (a-R,a+R)\) centered at \(a\), for some radius \(R>0\), then \[|R_n(x)| \le M\frac{|x-a|^{n+1}}{(n+1)!} \le M\frac{R^{n+1}}{(n+1)!}\] for all \(x\) in the interval. This will be useful soon for determining where a function equals its Taylor series. (If you're curious about the proof of the Lagrange error bound, there are basically two common ways to prove it: the Lagrange mean-value form of the remainder, or the integral form of the remainder.).