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m25-hw6

# m25-hw6 - that a n n =1 and b n n =1 are sequences of real...

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Math 25: Advanced Calculus Fall 2010 Homework # 6 Due Date: Friday, November 5 Reading: Finish reading sections 2.2 and 2.4. Read sections 2.5-2.7. Complete the following problems: 6.1: Decide if each of the following sequences { a n } n =1 converges or diverges. If the sequence converges, state its limit. In either case, you must use the appropriate definition prove that the sequence converges to the desired limit or that the sequence diverges. (a). a n = 1 n 2 (b). a n = n 2 + n n 2 (c). a n = n 2 + n n (d). a n = cos( ) (e). a n = ( - 1) n n 6.2: Suppose { a n } n =1 is a sequence of real numbers with the following property: there exists some N N such that a n = a N for all n N . Prove that lim n →∞ a n = a N . 6.3: Problem 2.7.2 in the textbook 6.4: Decide if each of the following statements is true or false. If the statement is true, give a proof. If it is false, provide a counterexample. In each of these problems, assume
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Unformatted text preview: that { a n } n =1 and { b n } n =1 are sequences of real numbers. (a). If { a n } and { b n } are both bounded, then so is { a n + b n } . (b). If { a n } and { b n } are both unbounded, then so is { a n + b n } . (c). If { a n } and { b n } are both bounded, then so is { a n b n } . (d). If { a n } is bounded, then { a n } is convergent. (e). If { a n } and { b n } are both divergent, then so is { a n + b n } . 6.5: Suppose { a n } and { b n } are sequences of real numbers such that { a n } is convergent and { b n } diverges to innity. Prove that lim n a n b n = 0 . 1...
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