1120_13 - Week 13 Outline Taylor Series and Maclaurin...

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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 0 ( 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 (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 0 , if x = 0 . 1
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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 integer n with 0 n N , the Taylor polynomial of order n
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