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6 - MATH 138 Winter 2011 Assignment 6 Topics Sequences...

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MATH 138 Winter 2011 Assignment 6 Topics: Sequences, series. Due: 11 a.m. Friday, February 18. 1. Prove the following theorem in two ways: “If lim n →∞ | a n | = 0 , then lim n →∞ a n = 0 .” (a) Using the squeeze theorem for sequences. Proof: We have -| a n | ≤ a n ≤ | a n | . We have lim n →∞ -| a n | = - lim n →∞ | a n | = 0 = lim n →∞ | a n | so by the squeeze theorem lim n →∞ a n = 0 . (b) Using the definition of limits. Proof: Since lim n →∞ a n = 0 , we know that > 0 , there exists N such that if n > N , we have | a n | - 0 < . That last condition is of course equivalent to | a n | < so lim n →∞ a n = 0 . 2. Determine whether the sequence converges or diverges. If it converges, find the limit. (a) a n = ( n - 1)! ( n +1)! Answer: Note that a n = 1 n ( n +1) . Since the sequences { 1 n } and { 1 n +1 } converge both to 0 , their product a n also converges to 0 . (b) b n = ( n +1)! ( n - 1)! Answer: Note that b n = n ( n + 1) . We know that f ( x ) = x ( x + 1) , and that lim x →∞ f ( x ) = . So lim n →∞ b n = lim n →∞ f ( n ) = lim x →∞ f ( x ) = hence the sequence { b n } diverges .
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