3100_practice_problems2

3100_practice_problems2 - PRACTICE PROBLEMS FOR SECOND MATH...

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PRACTICE PROBLEMS FOR SECOND MATH 3100 MIDTERM General Comments : These are practice problems. You should regard each indi- vidual problem as being a plausible exam problem. The exam will be closed-book and calculators will not be permitted. I would urge you not to use calculators or software when doing the practice problems – that would not be such good practice. When in doubt, you should justify your answers by correct logical reasoning (brief explanations are often sufficient). 1) Let { a n } n =1 be a sequence of real numbers. a) Say what it means for the infinite series n =1 a n to converge. b) Say what it means for n =1 a n to absolutely converge. c) Say what is means for n =1 a n to nonabsolutely converge. 2) Let { S n } n =1 be any real sequence. Show that there exists a real sequence { a n } such that for all n N , S n = a 1 + . . . + a n . 3) Let { a n } n =1 and { b n } n =1 be two positive sequences such that lim n =1 a n b n = .
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