# 00test2 - Mathematical Economics Midterm#2 November 6 2000...

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Mathematical Economics Midterm #2, November 6, 2000 1. Does the sequence { 1 , 1 / 1 , 2 , 1 / 2 , 3 , 1 / 3 , 4 , 1 / 4 , 5 , 1 / 5 , . . . } converge? Does it have any convergent subse- quences? Answer: The sequence does not converge. Let { x n } denote the sequence. Suppose it converged to x . Then for n large enough, | x n - x | < 1. Then | x n - x n +1 | < | x n - x | + | x - x n +1 | < 2 for n large. But this doesn’t happen. For n > 3, | x n - x n +1 | > 2. Therefore the sequence cannot converge. The subsequence of even terms, { 1 / 1 , 1 / 2 , 1 / 3 , . . . } , converges to zero. 2. Consider the dynamical system x t +1 = 3 / 2 1 / 2 1 1 x t a ) Find the eigenvalues of the matrix. Answer: The eigenvalue equation is (3 / 2 - λ )(1 - λ ) - 1 / 2 = 0. Solving the quadratic yields λ = 2 and λ = 1 / 2. b ) Find the corresponding eigenvectors. Answer: We start with λ = 2 and refer to the matrix as A . ( A - λ ) v = - 1 / 2 1 / 2 1 - 1 v 1 v 2 = [ 0 ] . The vector (1 , 1) T is a solution. Similarly, (1 , - 2) T solves the corresponding equation for λ = 1 / 2. c ) Are there any non-zero x 0 so that x t 0 ?

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