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Unformatted text preview: x does the series converge (b) absolutely, (c) conditionally? Denition 0.6 (Taylor and Maclaurin series) Let f be a function with derivatives of all orders through-out some interval containing a as an interior point. Then Taylor series generated by f at x = a is X k =0 f ( k ) ( a ) k ! ( x-a ) k (2) The Maclaurin series generated by f is X k =0 f ( k ) (0) k ! x k , the Taylor series generated by f at x = 0 . 1 Denition 0.7 (Taylor Polynomial of order n ) displaystyleP n ( x ) = n k =0 f ( k ) ( a ) k ! ( x-a ) k 6. Find the Taylor polynomials of orders , 1 , 2 and 3 . a. f ( x ) = 1 x , a = 2 b. f ( x ) = sin x, a = 4 7. Find the Malaurin series for a. e-x b. sin 3 x 8. Find the Taylor series a. f ( x ) = 1 x 2 , a = 1 b. f ( x ) = x 1-x , a = 0 2...
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