m25-hw7 - } converge and a n < b n for all n , then lim...

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Math 25: Advanced Calculus Fall 2010 Homework # 7 Due Date: Friday, November 12, 2010 Reading: Finish reading sections 2.5-2.7. Read sections 2.8-2.9. Complete the following problems: 7.1: Prove that lim n →∞ ( n + 1 - n ) = 0. [Hint: What would you have done in calculus?] 7.2: Decide whether or not each of the following statements is true or false. In either case, you must justify your answer with a proof. For all of these scenarios, assume that { a n } and { b n } are sequences of real numbers. (a). If { a n } is unbounded, then {| a n |} diverges to infinity. (b). If {| a n |} converges, then so does { a n } . (c). If { a n } and { b n } diverge, then so does { a n + b n } . (d). If { a n } and { a n + b n } converge, then so does { b n } . (e). If { a n } and { a n b n } converge, then so does { b n } . (f). If { a n } and { b n
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Unformatted text preview: } converge and a n < b n for all n , then lim n →∞ a n < lim n →∞ b n . 7.3: Prove that the sequence a n = 1 · 3 · 5 · ... · (2 n-1) 2 · 4 · 6 · ... · (2 n ) converges. [Note: The problem does not ask you to say what the limit of the sequence is!] 7.4: Suppose { a n } and { b n } are sequences of real numbers and a n ≤ b n for all n ∈ N . Prove that if a n diverges to ∞ , then b n diverges to ∞ as well. 7.5: Suppose { a n } is a nondecreasing sequence. Prove that 1. If { a n } is bounded, then lim n →∞ a n = inf { a n } . 2. If { a n } is unbounded, then lim n →∞ a n =-∞ . 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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