Expii

# 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.).