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].
Weierstrass Approximation Theorem
Taylor Series with Remainder