Let be continuous on a real interval containing (and ), and let exist at and be continuous for all . Then we have the following Taylor series expansion:
where is called the remainder term. Then Taylor's theorem [65, pp. 95-96] provides that there exists some between and such that
In particular, if in , then
which is normally small when is close to .
When , the Taylor series reduces to what is called a Maclaurin series [58, p. 96].