m25-hw9 - { a n } is an unbounded sequence of positive...

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Math 25: Advanced Calculus Fall 2010 Homework # 9 Due Date: Monday , November 29, 2010 Reading: Sections 2.10–2.13. Complete the following problems: 9.1: Let { a n } n =1 be a sequence of numbers, and let { a n k } k =1 be a subsequence. (a). If lim n →∞ a n = L , prove that lim k →∞ a n k = L as well. (b). If { a n } diverges to infinity, prove that { a n k } also diverges to infinity. 9.2: Show that each of the following sequences { t n } is a subsequence of either the sequence a n = n or the sequence b n = 1 n . Use Problem 9.1 to find lim n →∞ t n in each case. (a). t n = 1 n 3 (b). t n = n ! 9.3: Without simply appealing to Theorem 2.39, prove that if a sequence
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Unformatted text preview: { a n } is an unbounded sequence of positive numbers, then { a n } has a subsequence that diverges to infinity. 9.4: Consider the sequence defined by setting a = 2 and a n +1 = a n-a 2 n-2 2 a n . What is lim n →∞ a n ? 9.5: Using only the definition, prove that the sequence a n = 1 n is a Cauchy sequence. 9.6: Problem 2.13.2 (a,b,c,e) 9.7: Problem 2.13.9 9.8: Prove that lim sup n →∞ a n = ∞ if and only if { a n } has no upper bound. 1...
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This note was uploaded on 03/18/2012 for the course MAT MAT 25 taught by Professor Stevenklee during the Fall '10 term at UC Davis.

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