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Unformatted text preview: Exam I Solutions–Math 153 1:30pm, February 4, 2009 Instructor: Dr. Tseng NAME: NO CALCULATORS, NO BOOKS, NO SUPPLEMENTAL MATERIALS OF ANY KIND SHOW AND JUSTIFY ALL WORK (1) (15 points) If the sequence { a n } ∞ n =1 , where a n = ln( n ) sin( πn +3 2 n ) n 3 , converges, find its limit. If it diverges, explain why. Justify your answer completely. Solution: It converges. By L’Hopital’s rule lim n →∞ ln n n = 0. Also, by continu ity of the sine function, lim n →∞ sin( πn +3 2 n ) = sin( π/ 2) = 1 . Apply the product rule and power rule (with p = 3) for limits of convergent sequences (page 705 textbook), to finish. The limit equals 0. (2) (15 points) If the sequence { a n } ∞ n =1 , where a n = 3 sin 2 ( πn 2 ) + 1 , converges, find its limit. If it diverges, explain why. Justify your answer completely. Solution: It diverges. Notice that the even terms of the sequence are all 1 and the odd terms are all 4. Since this sequence does not have a unique limit, it diverges by definition. (3) (20 points) Determine whether the infinite series ∞ X n =1 2 sin( 1 3 n ) √ n converges or diverges. Be sure to justify your answer completely.converges or diverges....
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This note was uploaded on 03/29/2010 for the course MATH 153 taught by Professor Rempe during the Fall '08 term at Ohio State.
 Fall '08
 REMPE
 Math, Calculus

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