Taylor Series

# Forrest2 chapter 4 sequences and series 45

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Unformatted text preview: ll repeat the previous calculations for f (x) = sin(x). Example. Find all of the Taylor polynomials up to degree 5 for the function f (x) = sin(x) with center x = 0. We can easily see that f (0) = sin(0) = 0, f (0) = cos(0) = 0, f (0) = − sin(0) = 0, f (0) = − cos(0) = −1, f (4) (x) = sin(0) = 0, and f (5) (x) = cos(0) = 1. It follows that T0,0 (x) = 0, and T1,0 (x) = L0 (x) = 0 + 1(x − 0) = x DE Math 128 320 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series and 0 T2,0 = 0 + 1(x − 0) + (x − 0)2 2! =x = T1,0 (x) We get T3,0 (x) = 0 + 1(x − 0) + = x− 0 −1 (x − 0)2 + (x − 0)3 2! 3! x3 6 and that T4,0 (x) = 0 + 1(x − 0) + 0 −1 0 (x − 0)2 + (x − 0)3 + (x − 0)4 2! 3! 4! x3 6 = T3,0 (x) = x− Finally, we have T5,0 (x) = 0 + 1(x − 0) + 0 −1 0 1 (x − 0)2 + (x − 0)3 + (x − 0)4 + (x − 0)5 2! 3! 4! 5! x3 x5 + 5 5! x3 x5 = 1− + 5 120 = 1− The diagram below includes the graph of sin(x) along with all of its Taylor polynomials up to degree 5, though we have excluded T0,0 (x) since its graph is the x-axis. DE Math 128 321 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and...
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