Glossary

Taylor polynomial

The local polynomial approximation of a function built from its derivatives at a point.

Definition

It is Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k, and the Lagrange remainder Rn=f(n+1)(ξ)(n+1)!(x−a)n+1R_n=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} measures what is missing. Almost every differentiation, integration and ODE formula comes from truncating this expansion.

How it is used

It is the standard tool for deriving formulas and computing their order: the first neglected term dictates the hph^p power of the error.