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Unformatted text preview: Week 13 Outline Taylor Series and Maclaurin Series Generated by a Function –How to find the Taylor Series of a function in general –Some Useful Taylor Series for 1 1- x , e x , sin x Convergence of Taylor Series –Taylor’s Theorem –Remainder Estimation Theorem More Examples Applications of Taylor Series –Approximating the Values of Functions –Evaluating Nonelementary Integrals –Evaluating Indeterminate Forms Definition Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f at x = a is ∞ X k =0 f ( k ) ( a ) k ! ( x- a ) k = f ( a ) + f ( a )( x- a ) + f 00 ( a ) 2! ( x- a ) 2 + ··· + f ( n ) ( a ) n ! ( x- a ) n + ··· The Maclaurin series generated by f is ∞ X k =0 f ( k ) (0) k ! x k = f (0) + f (0) x + f 00 (0) 2! x 2 + ··· + f ( n ) (0) n ! x n + ··· , the Taylor series generated by f at x = 0. Remark If two power series ∑ a n ( x- a ) n and ∑ b n ( x- a ) n are convergent and equal for all values of x in an open interval containing a , then a n = b n for every n . Thus the Taylor series generated by f is unique. In particular, if f has a power series expansion at a , then it must be of the form f ( x ) = ∞ X k =0 f ( k ) ( a ) k ! ( x- a ) k . But there exist functions that are not equal to the sum of their Taylor series. For example, f ( x ) = ( e- 1 x 2 , if x 6 = 0 , if x = 0 . 1 Direct computation shows that the Maclaurin series of f is 0. Examples Find the Taylor Series of the following functions. 1. 1 1- x 2. e x 3. sin x 4. cos x 5. ln(1 + x ) 6. tan- 1 x Definition Let f be a function with derivatives of order k for k = 1 , 2 , ··· ,N in some interval containing a as an interior point. Then for any integeras an interior point....
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