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sec11_10 - 11.10 Taylor and Maclaurin Series 1 Taylor and...

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11.10 Taylor and Maclaurin Series 1. Taylor and Maclaurin Series Question Which functions have power series representations? How can we find such representations? Suppose f is a function that can be represented by a power series f ( x ) = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + c 3 ( x - a ) 3 + ... What are the coefficients c n in terms of f ? Theorem 1.1. If f has the power series representation at a , that is, if f ( 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 ! 1

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Section 11.10 Taylor and Maclaurin Series 2 Definition 1.1. The Taylor Series of a function f centered at a is the power series expansion of f ( x ) about a and is in the form f ( x ) = n =0 f ( n ) ( a ) n ! ( x - a ) n = f ( a ) + f ( a ) 1! ( x - a ) + f ( a ) 2! ( x - a ) 2 + f ( a ) 3! ( x - a ) 3 + · · · Definition 1.2. The Maclaurin Series of a function f is the power series expansion of f ( x ) about 0 and is in the form f ( x ) = n =0 f ( n ) (0) n ! x n = f (0) + f (0) 1! x + f (0) 2! x 2 + f (0) 3! x 3 + · · · Example 1.1. (problem 14) Find the Taylor series for f ( x ) = x - x 3 centered at x = - 2 .
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