5615H_MT2 - Math 5615H Name(Print 2nd Midterm Exam 60...

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Math 5615H. November 16, 2011. 2nd Midterm Exam. Name (Print) 60 points are distributed between 5 problems. You have 50 minutes (2:30 pm – 3:20 pm) to work on these problems. No books, no notes, except for: Appendix A. Exponential and Logarithmic Functions (3 pages). Calculators are permitted, however, for full credit, you need to show step- by-step calculations. This booklet consists of 7 page: the title page (this one), 5 pages (pp. 2–6) contain 5 problems. The almost empty page 7 can also be used for solutions. Some facts: 1. The number e = lim n →∞ 1 + 1 n · n = X n =0 1 n ! = 2 . 71828 . . . . 2. y = ln x for x > 0 ⇐⇒ x = e y ; 1 n + 1 < ln 1 + 1 n · < 1 n for n = 1 , 2 , 3 , . . . . 3. X n =1 1 n p converges ⇐⇒ p > 1 . 4. If | a n | ≤ c n for n = 1 , 2 , 3 , . . . , then X n =1 c n converges = X n =1 a n converges .
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Math 5615H. November 16, 2011. 2nd Midterm Exam. Page 2. Problem 1. (10 points). Let A 1 , A 2 , A 3 , . . . be subsets of a metric space ( X, d ). If B := [ k =1 A k , prove that B [ k =1 A k , where B and A k denote the closures of B and A k . Show, by an example, that this inclusion can be proper.
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Math 5615H. November 16, 2011. 2nd Midterm Exam. Page 3. Problem 2. (12 points). The sequence { a n } is defined by a 1 = 0 , and a n +1 = 2 a n / 2 for n = 1 , 2 , 3 , . . . . Prove that { a n } is convergent and find its limit. You can use without proof
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