midterm1 - Midterm Exam 1 Math 451, Prof. Roman Vershynin...

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Unformatted text preview: Midterm Exam 1 Math 451, Prof. Roman Vershynin Fall 2011 Name: Read the following information before starting the exam: • No laptops or any communication devices are allowed on the exam. • Show all work, clearly and in order, if you want to get full credit. Points may be taken off if it is not clear how you arrived at your answer (even if your final answer is correct). • Please keep your written answers brief; be clear and to the point. Points may be taken off for rambling and for incorrect or irrelevant statements. Problem Points 1 2 3 4 5 6 (bonus) Total 1. ( 10 points ) For each of the following sequences, compute inf { s n } , sup { s n } , lim inf s n and lim sup s n . No justification is necessary; you may just write down the answers. a. ( 5 pts ) s n = (- 1) n n + 1 + (- 1) n 2 b. ( 5 pts ) s n =- n 2 + (- 1) n 2. ( 20 points ) Let ( s n ) be a convergent sequence and let lim s n = s . Show that ( | s n | ) is a convergent sequence and lim | s n | = | s | . Use the definition of limit in your argument. 3.3....
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This note was uploaded on 12/13/2011 for the course MATH 451 taught by Professor Staff during the Fall '08 term at University of Michigan.

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midterm1 - Midterm Exam 1 Math 451, Prof. Roman Vershynin...

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