Taylor Series and Power Series from Old Exams

# Thomas' Calculus: Early Transcendentals

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Mat104 Taylor Series and Power Series from Old Exams (1) Use MacLaurin polynomials to evaluate the following limits: (a) lim x 0 e x - e - x - 2 x x 2 - x ln(1 + x ) (b) lim x 0 cos( x 2 ) - 1 + x 4 / 2 x 2 ( x - sin x ) 2 (2) Evaluate or show that lim n →∞ n tan 1 n (3) Find lim x 0 sin x · e x 2 - x ln(1 + x 3 ) . (4) Find lim n →∞ n 2 1 - cos 1 n or show that it does not exist. (5) Find lim x 0 1 sin x - 1 1 - e - x (6) Find lim x 0 cos( x 3 ) - 1 sin( x 2 ) - x 2 . (7) Find lim x 0 sin x - x (cos x - 1)( e 2 x - 1) . (8) For what real values of x does n =0 ( - 1) n x n n 2 + 1 converge? Give your reasons. (9) For what real values of x does n =1 e n ( x - 1) n 2 n · n converge? Give your reasons. (10) Let f ( x ) = x 0 sin( t 2 ) dt . Find the Taylor series at 0 (i.e., the Taylor series about a = 0) of f ( x ), giving enough terms to make the pattern clear. Also, find the 100th derivative of f ( x ) at 0. (11) Find the Taylor series at 0 of f ( x ) = 1 - cos(2 x 2 ) x and find f (7) (0) and f (8) (0). (12) Find the Taylor series about 0 of (a) ln(1 + x 3 ) (b) x 1 + x 2 (13) (a) Find the Taylor series at x = 0 for e x 2 .

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• Fall '07
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