## Formal Statement of Taylor's Theorem

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 [63, 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 [56, p. 96].

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