## 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

When , the Taylor series reduces to what is called a *Maclaurin
series* [56, p. 96].

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Taylor Series with Remainder