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 n1 = 7 n n ! · ( n1)! 7 n1 = 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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 Spring '09
 CHRIST
 Squeeze Theorem, Lecturer, positive integer values, Prof. Mina Aganagic, Mina Aganagic GSI

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