Expii

# 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 $$I$$.).