hw6 - IE 500 Fall 2009 E. Alper Yldrm HOMEWORK 6 (due...

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IE 500 – Fall 2009 E. Alper Yıldırım HOMEWORK 6 (due Wednesday, December 23 in class) 1. Let ( x n ) and ( y n ) be two sequences of real numbers. (a) Suppose that i =1 y i is convergent. Suppose also that | x n | ≤ y n for each n N . Prove that i =1 x i is a convergent series. (b) Suppose that y n x n 0 for each n N . Suppose that i =1 x i is NOT a convergent series. Prove that i =1 y i is not a convergent series. 2. (a) Let ( x n ) be a sequence of nonnegative real numbers such that x 1 x 2 ... 0. Consider the sequence given by y n = 2 n - 1 x 2 n - 1 , n N (i.e., y 1 = x 1 ,y 2 = 2 x 2 ,y 3 = 4 x 4 ,y 4 = 8 x 8 , etc.). Suppose that i =1 y i is convergent. Prove that i =1 x i is convergent. (b) Suppose that x n = 1 /n q for n N , where q R . Prove, using the previous part, that the series i =1 x i is convergent for q > 1. 3. (a) Let ( X,d ) be a metric space. Let D : X × X [0 , ) be defined as D ( x,y ) = ± 0 if x = y, max { 1 ,d ( x,y ) } otherwise.
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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