pra-140 - Math 140, Winter Instructor: Professor Ni...

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Math 140, Winter Practice problems February, 2010 Instructor: Professor Ni 1. Prove that if x is a real number and | x | < 1, then lim n →∞ x n = 0. 2. The Cantor set are constructed as following. Let K 0 = [0 , 1], remove ( 1 3 , 2 3 ) let K 1 = [0 , 1 3 ] [ 2 3 , 1]. Then remove the middle thirds of these intervals and let K 2 = [0 , 1 9 ] [ 2 9 , 3 9 ] [ 6 9 , 7 9 ] [ 8 9 , 1]. Continue this process and construct K n . Then P = n =1 K n . Prove that 1) P is nonempty and compact. 2) P is a perfect set. 3. Let X be a metric space in which every infinite subset has a limit point. Prove that X is separable. 4. Define e = X n =1 1 n ! . Prove that 1) e = lim n →∞ 1 + 1 n · n . 2) e is irrational. 5. Find the convergence radius of (i) n =1 n ! z n , (ii) n =1 q n 2 z n , | q | < 1, (iii) n =1 z n ! . 6. Suppose that a n 0 for all n 1 and n =1 a n diverges. Prove that 1) n =1 a n 1+ a n and n =1 a
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This note was uploaded on 09/30/2011 for the course MATH 140a taught by Professor Staff during the Fall '08 term at UCSD.

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pra-140 - Math 140, Winter Instructor: Professor Ni...

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