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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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 Fall '07
 Nelson
 Polynomials, Power Series, Taylor Series, Limits

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