Taylor Series

# 1 1 and x a0 1 we have the following f a x a0 0 f

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Unformatted text preview: it by T3,a (x). Given a function f (x), we will also write T0,a (x) = f (a) and T1,a (x) = La (x) = f (a) + f (a)(x − a) and call these polynomials the zero-th degree and ﬁrst degree Taylor polynomials of f (x) centered at x = a, respectively. Observe that with the convention that 0! = 1! = 1 and (x − a)0 = 1 , we have the following f (a) (x − a)0 0! f (a) f (a) T1,a (x) = (x − a)0 + (x − a)1 0! 1! f (a) f (a) f (a) T2,a (x) = (x − a)0 + (x − a)1 + (x − a)2 0! 1! 2! f (a) f (a) f (a) f (a) T3,a (x) = (x − a)0 + (x − a)1 + (x − a)2 + (x − a)3 0! 1! 2! 3! T0,a (x) = Recall that f (k) (a) denotes the k -th derivative of f (x) at x = a and that by convention f (0) (x) = f (x). Then using summation notation, we get 0 T0,a (x) = k=0 1 T1,a (x) = k=0 2 T2,a (x) = k=0 f k (a) (x − a)k k! f k (a) (x − a)k k! f k (a) (x − a)k k! and 3 T3,a (x) = k=0 f k (a) (x − a)k k! This leads us to the following deﬁnition: Deﬁnition. Assume that f (x) is n-times diﬀerentiable at x = a. The n-th degree Taylor polynomial for f (x) centered at...
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## This document was uploaded on 03/23/2013.

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