# Answer it does not converge note that x n x n 1 2 b

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Answer: It does not converge. Note that | x n - x n +1 | 2. b ) If it doesn’t converge, does it have any convergent subsequences? If so, identify one of them and compute its limit. Answer: Yes, it has convergent subsequences. Two obvious ones are the sub- sequence of even terms (with limit 1) and the subsequence of odd terms (with limit - 1). 3. Consider the differential system d dt x y = 3 x + y x + 3 y with initial conditions x (0) = 2 and y (0) = 3. a ) Write the system in matrix form. Answer: d dt x y = 3 1 1 3 x y .

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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 6, 2003 Page 2 b ) Find the eigenvalues of this system. Answer: The eigenvalue equation is 0 = (3 - λ )(3 - λ ) - 1 = λ 2 - 6 λ + 8. This has solutions λ = 2 and λ = 4. c ) Is this system stable? Answer: No, both eigenvalues are positive so it is unstable. d ) Find the solution to this system. Answer: Since the original matrix is symmetric, we can find orthonormal eigen- values. One eigenvector corresponding to λ = 4 is (1 , 1) 0 and an eigenvector corresponding to λ = 2 is (1 , - 1) 0 . In both cases we can normalize by dividing by 2 to obtain orthonormal eigenvectors.
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