Extension: Integral 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 integral form, is \[ R_n(x) = \frac{1}{n!}\int_a^x (x-t)^n f^{(n+1)}(t)\, dt.\] (Technical hypotheses: this formula holds if \(f^{(n+1)}\) is continuous on an open interval \(I\) containing \(a\), and \(x\) lies in \(I\).).