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Exam_solutions_4_(2)

Exam_solutions_4_(2) - MATH 141 TEST 4(9.1 9.9[Pilachowski...

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Math 141, Sections 03**, T Pilachowski, Spring 2007 April 27, 2007 MATH 141 – TEST 4 (9.1 – 9.9) [Pilachowski] 1. a. (12 points) Use derivatives to find the 4th Taylor polynomial ( ) x p 4 about x = 0 for the function ( ) x e x f 2 = . (You must show your steps to receive full credit.) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 16 0 16 8 0 8 4 0 4 2 0 2 1 0 4 2 4 3 2 3 2 2 2 1 2 1 2 = = = = = = = = = = f e x f f e x f f e x f f e x f f e x f x x x x x ( ) 4 3 2 4 3 2 4 3 2 3 4 2 2 1 4 16 3 8 2 4 2 1 ! ! ! x x x x x x x x x p + + + + = + + + + = b. (10 points) Find the sum of the series = + + 0 3 2 1 2 3 n n n . ( ) 2 3 4 8 3 1 8 3 1 8 3 4 3 8 3 2 3 8 3 2 3 2 3 2 3 4 1 4 3 0 4 3 0 0 2 0 2 3 0 3 2 1 = = = = = = = = = = = + + n n n n n n n n n n 2. a. (12 points) Determine whether the series = + 1 5 2 n n n converges. (You must show work that supports your answer.) 0 2 1 5 2 lim = + n n n , so the series diverges; Other methods, correctly used, are acceptable.
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