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quiz 4.2

# quiz 4.2 - → ∞ So by the Squeeze Theorem lim x →∞ f...

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Math 1B: Calculus February 24, 2010 Quiz 4 Lecturer: Prof. Mina Aganagic GSI: Gary Sivek Name: Answers Determine whether the following sequences converge or diverge. If they converge, find the corresponding limit. (Justify your answer.) 1. (2 pts) a n = cos 3 n log n + 1 /n To be perfectly rigorous, we will define a function f ( x ) = cos 3 x log x +1 /x which agrees with the sequence { a n } at positive integer values of x . Then if f ( x ) has a limit, the sequence will have the same limit. Now | cos 3 x | 1 for all x , since cos is bounded. Note that this means we will not be able to apply L’Hopital’s rule! However, we do have for x 1: - 1 log x + 1 /x f ( x ) 1 log x + 1 /x and both the left and right sides have limit 0 as
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Unformatted text preview: → ∞ . So by the Squeeze Theorem, lim x →∞ f ( x ) = 0, and thus lim n →∞ a n = 0 as well. 2. (3 pts) b n = 7 n n ! There are many ways to show that this approaches 0. As one possibility, observe that whatever b 13 is, it is positive, and that for n ≥ 14 we have: b n b n-1 = 7 n n ! · ( n-1)! 7 n-1 = 7 n ≤ 1 2 So b 14 ≤ (1 / 2) b 13 , and b 15 ≤ (1 / 2) b 14 ≤ (1 / 2) 2 b 13 , and b 16 ≤ (1 / 2) b 15 ≤ (1 / 2) 3 b 13 , and so on. In general, for any positive n we have 0 ≤ b 13+ n ≤ (1 / 2) n b 13 , and we may apply the Squeeze Theorem to show that lim n →∞ b n = 0....
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