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exam1solutions - Calculus III Math 143 Spring 2008...

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Calculus III Math 143 Spring 2008 Professor Ben Richert Exam 1 Solutions Problem 1. (20pts) Consider the sequence 2 + n n n =1 . (a–10pts) Does this sequence converge or diverge? Solution. Note that the function f ( x ) = 2 + x x is continuous on [1 , ) and of course f ( n ) = 2 + n n , so to show that the sequence converges, we only need to show that lim x →∞ f ( x ) exists. Of course, the form of lim x →∞ f ( x ) is “ ”, and the derivative of the top and bottom exist and are nonzero, so by L’Hospital’s Rule, lim x →∞ 2 + x x = lim x →∞ 1 2 x 1 = lim x →∞ 1 2 x = 0 . We conclude that the sequence converges. (b–10pts) Does the series X n =1 2 + n n converge or diverge? Solution. Note that 0 1 n 2 n 2 + n n for all n 1 since n 0. Thus by the comparison test, it is enough to show that X n =1 1 n diverges. But this follows immediately from the P test, and we are finished. Problem 2. (10pts) Do one of the following two problems. (a–10pts) Decide if X n =1 ( - 1) n n e n converges or diverges. Solution. Since n e n is always positive, the series clearly alternates. By the alternating series test, we will know the series converges if n n e n o 0 and
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