10.4 | The Error in Taylor Polynomial Approximations


Suppose \(f\) and all its derivatives are continuous. If \(P_n(x)\) is the \(n^{th}\) Taylor polynomial for \(f(x)\) about \(a\), then
$$
|E_n(x)| = |f(x) – P_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1},
$$
where \(|f^{n+1}(x)| \leq M \) on the interval between \(a\) and \(x\).

E 10.4 Exercises