508F10Ex1s

# 508F10Ex1s - B–2 Show that 1 1 2 1 3 1 4 1 k< 2 B–3...

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Math 508 Exam 1 Jerry L. Kazdan October 14, 2010 9:00 – 10:20 Directions This exam has three parts, Part A asks for 4 examples (20 points, 5 points each). Part B has 4 shorter problems (36 points, 9 points each. Part C has 3 traditional problems (45 points, 15 points each). Total is 101 points. Closed book, no calculators or computers– but you may use one 3 ±± × 5 ±± card with notes on both sides. Part A: Examples (4 problems, 5 points each so 20 points). Give an example having the speciﬁed property. 1. A subset of the real line R that contains all but three of its limit points. 2. A sequence of real numbers x n with the property that | x n +1 - x n | → 0 but the sequence does not converge. 3. A collection of closed intervals A k R , k = 1 , 2 . . . , whose union is not closed. 4. A collection of bounded nested intervals J 1 J 2 J 3 ⊇ ··· whose intersection is empty. Part B: Short Problems (4 problems, 9 points each so 36 points) B–1. Let a k = k + 1 - k , k = 1 , 2 , . . . . Determine if this sequence converges.

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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 +1-a n | ≤ c | a n-a n-1 | , n = 1 , 2 , 3 , . . . . a) Show that | a n +1-a n | ≤ c n | a 1-a | . b) If n > k , show that | a n-a k | ≤ c k 1-c | a 1-a | . c) Show that the sequence a n converges. 2...
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508F10Ex1s - B–2 Show that 1 1 2 1 3 1 4 1 k< 2 B–3...

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