Taylor Series and Power Series from Old Exams

Thomas' Calculus: Early Transcendentals

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Unformatted text preview: Mat104 Taylor Series and Power Series from Old Exams (1) Use MacLaurin polynomials to evaluate the following limits: (a) lim x e x- e- x- 2 x x 2- x ln(1 + x ) (b) lim x 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 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 1 sin x- 1 1- e- x (6) Find lim x cos( x 3 )- 1 sin( x 2 )- x 2 . (7) Find lim x sin x- x (cos x- 1)( e 2 x- 1) . (8) For what real values of x does X n =0 (- 1) n x n n 2 + 1 converge? Give your reasons. (9) For what real values of x does X n =1 e n ( x- 1) n 2 n n converge? Give your reasons. (10) Let f ( x ) = Z x 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....
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Taylor Series and Power Series from Old Exams - Mat104...

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