quiz 4.2 - . So by the Squeeze Theorem, lim x f ( x ) = 0,...

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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, ±nd the corresponding limit. (Justify your answer.) 1. (2 pts) a n = cos 3 n log n +1 /n To be perfectly rigorous, we will de±ne a function f ( x )= cos3 x log x +1 /x which agrees with the sequence { a n } at positive integer values of x .T h e ni f f ( x )ha sal im i t , the sequence will have the same limit. Now | cos 3 x |≤ 1fora l l x ,s incecosisbounded .Notethatth ismeanswew i l l not be able to apply L’Hopital’s rule! However, we do have for x 1: - 1
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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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This note was uploaded on 10/20/2010 for the course MATH 53903 taught by Professor Christ during the Spring '09 term at University of California, Berkeley.

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