Taylor Series

# The following theorem provides us with the answer to

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Unformatted text preview: d version of the Mean Value Theorem, called Taylor’s Theorem. We begin with some useful notation. Deﬁnition. Assume that f (x) is n times diﬀerentiable at x = a. Let Rn,a (x) = f (x) − Tn,a (x) Rn,a (x) is called the n-th degree Taylor remainder function centered at at x = a. The error in using the Taylor polynomial to approximate f (x) is given by Error =| Rn,a (x) | The central problem for this approximation process is: Problem: Given a function f (x) and a point x = a, how do we estimate the size of Rn,a (x)? The following theorem provides us with the answer to this question: Theorem. [Taylor’s Theorem] Assume that f (x) is n + 1-times diﬀerentiable on an interval I containing x = a. Let x ∈ I . Then there exists a point c between x and a such that f (x) − Tn,a (x) = Rn,a (x) = f (n+1) (c) (x − a)n+1 . (n + 1)! We will make three important observations about this theorem. First, since T1,a (x) = La (x), when n = 1 the absolute value of the remainder Rn,a (x) represents the error in using the...
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