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Spring 2006 - Zabrocki's Class - Exam 2 (Version A)

Spring 2006 - Zabrocki's Class - Exam 2 (Version A) - score...

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Math 20D - Spring 2006 - Midterm 2- Prof. Mike Zabrocki score 1 24 2 25 3 25 4 26 BONUS 10 total 100 Name: Time of section: 1. Determine if the following sums diverge, converge absolutely or converge conditionally. State which test you use to justify your answer and the conclusion which follows from that test. (a) n =1 sin( 1 n 2 ) . (b) n =2 ( - 1) n n - n 2 Since sin( 1 n 2 ) = 1 n 2 - ( 1 n 2 ) 3 3! + · · · , This is an alternating series and 1 n 2 - n we know that lim n →∞ sin ( 1 n 2 ) 1 n 2 = 1 is positive and decreasing and lim n →∞ 1 n 2 - n = 0 and by limit comparison test we know that Hence by A.S.T. converges conditionally. n 0 1 n 2 converges since lim n →∞ 1 n 2 - n 1 n 2 = 1 implies that n 1 sin( 1 n 2 ) converges. then this series converges absolutely The series is positive hence converges by the L.C.T. with 1 n 2 absolutely (c) n =1 sin( πn 2 ) 1+sin( n 2 ) (d) n =1 (1 - e 1 /n ) lim n →∞ 1 1+sin( n 2 ) = 0 Since 1 - e 1 n = 1 - (1 + 1 n + ( 1 n ) 2 2!
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