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Taylor Series

# That is tna encodes not only the value of f x at x a

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Unformatted text preview: x = a is the polynomial DE Math 128 318 (B. Forrest)2 CHAPTER 4. Sequences and Series n Tn,a (x) = k=0 4.5. Introduction to Taylor Series f k (a) (x − a)k k! = f (a) + f (a)(x − a) + f (n) (a) f (a) (x − a)2 + · · · + (x − a)n 2! n! The remarkable thing about Tn,a (x) is that for any k between 0 and n, (k ) Tn,a (a) = f (k) (a). That is Tn,a encodes not only the value of f (x) at x = a but all of its ﬁrst n derivatives as well. Moreover, this is the only polynomial of degree n or less that does so. Example. Find all of the Taylor polynomials up to degree 5 for the function f (x) = cos(x) with center x = 0. We have already seen that f (0) = cos(0) = 1, f (0) = − sin(0) = 0 and f (0) = − cos(0) = −1. It follows that T0,0 (x) = 1, and T1,0 (x) = L0 (x) = 1 + 0(x − 0) = 1 for all x, while T2,0 = 1 + 0(x − 0) + x2 −1 (x − 0)2 = 1 − . 2! 2 Since f (x) = sin(x), f (4) (x) = cos(x), and f (5) (x) = − sin(x), we get f (0) = sin(0) = 0, f (4) (0) = cos(0) = 1 and f (5) (x) = − sin(0) =...
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