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Unformatted text preview: B–2. Show that 1 + 1 2! + 1 3! + 1 4! + . . . + 1 k ! + . . . < 2 B–3. Given any two rational numbers p < q , prove there is an irrational number c between them. B–4. Given any c > 1, show that c n n ! → 0. Part C: Traditional Problems (3 problems, 15 points each so 45 points) C–1. Let x k ≥ 0 be a sequence of real numbers that converges to A . Show that A ≥ 0. C–2. Given a sequence { a k } of real numbers, let S n = a 1 + ··· + a n n be the sequence of averages ( arithmetic mean ). If a k converges to 0, show that the averages S n also converge to 0. C–3. Let { a n } ∈ C be a contracting sequence, that is there is a 0 < c < 1 so that  a n +1a n  ≤ c  a na n1  , n = 1 , 2 , 3 , . . . . a) Show that  a n +1a n  ≤ c n  a 1a  . b) If n > k , show that  a na k  ≤ c k 1c  a 1a  . c) Show that the sequence a n converges. 2...
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 Fall '10
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 Math, Rational number, Irrational number, Jerry L. Kazdan

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