# n_13492 - 11.10 Taylor and Maclaurin Series Find the Taylor...

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§ 11.10 Taylor and Maclaurin Series Find the Taylor series (or Maclaurin Series) of a function Calculate the coefficients directly by using Theorem 5 . Theorem 5 If f has a power series representation (expansion) at a , that is, if f ( x ) = X n =0 c n ( x - a ) n | x - a | < R then its coefficients are given by the formula c n = f ( n ) ( a ) n ! Use some Taylor series (or Maclaurin Series) we have already known. * Some Important Maclaurin Series. 1 1 - x = X n =0 x n = 1 + x + x 2 + x 3 + · · · ( - 1 , 1) e x = X n =0 x n n ! = 1 + x 1! + x 2 2! + x 3 3! + · · · ( -∞ , ) sin x = X n =0 ( - 1) n x 2 n +1 (2 n + 1)! = x - x 3 3! + x 5 5! - x 7 7! + · · · ( -∞ , ) cos x = X n =0 ( - 1) n x 2 n (2 n )! = 1 - x 2 2! + x 4 4! - x 6 6! + · · · ( -∞ , ) arctan x = X n =0 ( - 1) n x 2 n +1 2 n + 1 = x - x 3 3 + x 5 5 - x 7 7 + · · · [ - 1 , 1] * Multiplication and Division: the first few terms. 1

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Prove the Taylor series (or Maclaurin Series) of one function f(x) represents f(x) – Theorem 8 If f ( x ) = T n ( x )+ R n ( x ), where T n ( x ) is the n th-degree Taylor polynomial of f at a and lim n →∞ R n ( x ) = 0 for | x - a | < R , then f
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