SampleProblemsForSequenceAndSeries_Modified2_-1 - Integral...

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Integral Calculus MTH 62-141 Problems about Sequence and Series 1. Sequence (a) Determine whether the sequence is convergent, absolutely convergent, condition- ally convergent or divergent. If it converges, find the limit i. a n = n 3 n 3 +1 . Answer: 1 ii. a n = e 1 /n . Answer: 1 iii. a n = tan 2 1 + 8 n . Answer: 1 iv. a n = ( - 1) n - 1 n n 2 +1 . Answer: 0 v. a n = cos( n/ 2). Answer: diverges vi. a n = cos(2 /n ). Answer: 1 vii. a n = ln n ln 2 n . Answer: 1 viii. a n = e n + e - n e 2 n - 1 . Answer: 0 ix. a n = ( 1 + 2 n ) n . Answer: e 2 x. a n = sin 2 n 1+ n . Answer: 0 xi. a n = ln(2 n 2 + 1) - ln( n 2 + 1). Answer: ln 2 xii. { 0 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , · · · } . Answer diverges (b) Estimate the limit. Then use definition of limit to prove it. (for section 04 only) i. a n = 1 n . ii. a n = 1 n 2 . iii. a n = n n +1 . iv. a n = e - n sin( n ). v. a n = e - n . vi. a n = e n - e - n e n + e - n . vii. a n = sin( n ) n . (c) Determine whether the sequence is increasing, decreasing or not monotonic. Is the sequence bounded? i. a n = 1 2 n +3 . Answer: decreasing, bounded ii. a n = 2 n - 3 3 n +4 . Answer: increasing, bounded iii. a n = n ( - 1) n . Answer: not monotonic, not bounded iv. a n = ne - n . Answer: decreasing, bounded
v. a n = n + 1 n . Answer: increasing, bounded below 2. Series (a) Determine whether the series is convergent or divergent. If it is convergent, find its sum when it is possible. i. using definition of convergence A. X n =1 1 n ( n + 1) . Answer: converges, 1 B. X n =1 1 n - 1 n + 1 . Answer: converges, 1 C. X n =2 1 n 2 - 1 . Answer: converges, 3 4 D. X n =1 3 n ( n + 3) . Answer: converges, 11 6 E. X n =1 2 n 2 + 4 n + 3 . Answer: converges, 5 6 F. X n =1 ln n n + 1 . Answer: diverges G. X n =1 2 3 n - 1 . Answer: converges, 3 H. X n =1 n 2 3 n - 1 . [Hint: You may use X n =1 nx n - 1 = nx n +1 - ( n + 1) x n + 1 (1 - x ) 2 ] Answer: converges, 9 ii. geometric series A. 3 + 2 + 4 3 + 8 9 + · · · . Answer: converges, 9 B. X n =1 1 + 2 n 3 n . Answer: converges, 5 2 C. X n =1 1 + 3 n 2 n . Answer: diverges D. X n =1 (cos 1) n . Answer: converges, cos 1 1 - cos 1 iii. lim n →∞ a n does not exist or lim n →∞ a n 6 = 0 A.

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