taylor_series

taylor_series - (12.10) Taylor Series In this section, we...

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(12.10) Taylor Series In this section, we learn a general technique for Fnding power series that represent a given function. Suppose f ( x )= a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + ··· . (1) We want to Fgure out what the coefficients a n are, so that the power series does represent the function. If we let x = 0, then f (0) = a 0 + a 1 0+ a 2 0 2 + a 3 0 3 + a 4 0 4 = a 0 . So at least we know a 0 = f (0). Now take the derivative of equation (1). We get f ± ( x a 1 +2 a 2 x +3 a 3 x 2 +4 a 4 x 3 + . (2) Now let x = 0. We get f ± (0) = a 1 a 2 0+3 a 3 0 2 a 4 0 3 + = a 1 .So a 1 = f ± (0). Now take the derivative of equation (2). We get f ±± ( x )=2 a 2 +6 a 3 x +12 a 4 x 2 + . (3) Let x = 0. We get f (0) = 2 a 2 a 3 0+12 a 4 0 2 + =2 a 2 a 2 = f (0) 2 . Now take the derivative of equation (3). We get f ±±± ( x )=6 a 3 +24 a 4 x + . Let x = 0. We get f (0) = 6 a 3 ,so a 3 = f (0) 6 . Keep going with this. We get a n = f (0) (0) n ! . That is, if f ( x a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + = ± n =0 a n x n then a n = f (0) (0) n ! . We call this the Maclaurin series for f ( x ).
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This note was uploaded on 04/02/2008 for the course MATH 31B taught by Professor Valdimarsson during the Fall '08 term at UCLA.

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taylor_series - (12.10) Taylor Series In this section, we...

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