exam1solutions

Exam1solutions - Calculus III Math 143 Spring 2008 Professor Ben Richert Exam 1 Solutions Problem 1(20pts Consider the sequence 2 √ n n ∞ n

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Unformatted text preview: 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....
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This note was uploaded on 06/08/2008 for the course MATH 143 taught by Professor Staff during the Spring '03 term at Cal Poly.

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Exam1solutions - Calculus III Math 143 Spring 2008 Professor Ben Richert Exam 1 Solutions Problem 1(20pts Consider the sequence 2 √ n n ∞ n

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