Taylor Series

# N0 which is exactly the same taylor series as for ex

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Unformatted text preview: ntered at x = 0 for g (x) is ∞ xn n! n=0 which is exactly the same Taylor Series as for ex . However at x = 2, g (2) = e while ∞ n=0 2n = e2 = g (2). n! Example. Find the Taylor series centered at x = 0 for f (x) = cos(x). We have that f (x) f (x) f (x) f (4) (x) f (5) (x) f (6) (x) f (7) (x) f (8) (x) = sin(x) = − cos(x) = sin(x) = cos(x) = − sin(x) = − cos(x) = sin(x) = cos(x) DE Math 128 =⇒ =⇒ =⇒ =⇒ =⇒ =⇒ =⇒ =⇒ . . . f (0) f (0) f (0) f (4) (0) f (5) (0) f (6) (0) f (7) (0) f (8) (0) = − sin(0) = − cos(0) = sin(0) = cos(0) = − sin(0) = − cos(0) = sin(0) = cos(0) 311 =0 = −1 =0 =1 =0 = −1 =0 =1 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series with the cycle repeating itself every four derivatives. This gives f (4k) (x) f (4k+1) (x) f (4k+2) (x) f (4k+3) (x) = cos(x) = − sin(x) = − cos(x) = sin(x) f (4k) (0) f (4k+1) (0) f (4k+2) (0) f (4k+3) (0) =⇒ =⇒ =⇒ =⇒ . . . = cos(0) = − sin(0) = − cos(0) = sin(0) =1 =0 = −1...
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## This document was uploaded on 03/23/2013.

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