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solnsexamII - 1 SOLUTIONS(1(20 points(a If the sequence an...

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1. SOLUTIONS (1)(20 points) ( a ) If the sequence a n satisfies lim n →∞ a n +1 - a n = 0 then lim n →∞ a n exists and is finite. F Take a n = n. ( b ) If q is a polynomial of degree 5 and q (0) = q (0) = q (4) (0) = 0 and q (0) = 1 , q (0) = - 1 6 , q (5) (0) = 1 120 then q is the fifth degree Taylor polynomial for f ( x ) = sin x at c = 0 . T Since f (0) = f (0) = f (4) (0) = 0 and f (0) = 1 , f (0) = - 1 6 , f (5) (0) = 1 120 . ( c ) The matrix A = 0 1 0 0 0 1 0 0 0 is diagonalizable. F One way to see this is that the only eigenvalue is 0 , it has multiplicity 3 , and the eigenspace N ( A - 0 I ) = { ( t, 0 , 0) T : t R } so there are not three linearly independent eigenvectors. Another way to see this is that if A were diagonalizable, then for some invertible matrix X we would have A = XDX - 1 and D is the diagonal matrix with the eigenvalues of A on its diagonal. But this would be the 0 matrix and as a result A is also the 0 matrix, but it is not, so A can not be diagonalizable. 1
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2 ( d ) If an eigenvalue of the matrix A has multiplicity greater than one, then A is defective. F Defective means not diagonalizable, but the identity matrix has sole eigenvalue 1 and is already diagonal.
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