Math+2J+Practice+Final

# Math+2J+Practice+Final - Part 2 Sequence and Infinite...

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Math 2J Practice Final Note: this is only a practice final! It should not be your only study guide! Make sure you redo all the examples we did in class! Part 1: Linear Algebra 1. Diagonalize the matrix 4 4 8 4 6 4 6 4 10 A - - - = . 2. Solve the linear system 1 2 1 2 3 1 2 3 2 3 2 4 5 6 2 5 3 x x x x x x x x - = - - + = - - + = - by (a) Gaussian Elimination (b) Cramer’s rule 3. True or False (a) A homogeneous linear system can have infinitely many solutions (b) The product of two elementary matrices is an elementary matrix (c) An overdetermined system always has infinitely many solutions (d) det(A+B)=det(A)+det(B) (e) A triangular matrix is nonsingular if and only if the diagonal entries are all nonzero. (f) If λ is an eigenvalue of A, then 1 λ is an eigenvalue of A -1 . (g) The determinant of a matrix is the sum of its eigenvalues. (h) A matrix is diagonalizable only if all the eigenvalues are distinct. (i) If A and B are row equivalent then their determinants are equal. (j) If x is a nonzero vector and Ax = 0, then det(A)=0. 4. Redo all the problems on practice midterm.

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Unformatted text preview: Part 2: Sequence and Infinite Series 1. Find the interval of convergence for the following series (a) 1 2 n n n nx + = (b) ( 29 1 1 ! n n n n n x n =-(c) ( 29 ( 29 2 1 1 2 1 n n n n x n + =-+ (d) ( 29 1 1 10 ! n n n n x n + =-(e) ( 29 ( 29 1 2 1 n n n n x n =-+ (f) ( 29 2 1 4 1 n n n nx n = + 2. Determine the convergence or divergence of the following series (a) 3 1 3 2 n n n n = (b) ( 29 1 ! ! 2 ! n n n n = (c) ( 29 2 1 10 1 3 2 n n n n n = + + + (d) 1 cos 2 n n π = (e) ( 29 1 3 ! n n n =-(f) ( 29 2 cos n n n = 3. Find the Taylor series for (a) 2 1 ( ) about 1 f x c x = = -(b) ( ) ln about 2 f x x c = = 4. Determine whether the sequence converges or diverges. If converges, find the limit (a) { } 2 n a n n = +-(b) ( 29 2 3 1 n n + -(c) ! 2 n n n a = (d) 2 cos 2 n n n a = (e) ( 29 2 3 1 1 n n n a n = -+ (f) ( 29 1 n n-(g) 2 2 2 1 5 3 n n n + +...
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## This note was uploaded on 03/11/2012 for the course MATH 2J 44360 taught by Professor Donaldson,neil during the Spring '10 term at UC Irvine.

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Math+2J+Practice+Final - Part 2 Sequence and Infinite...

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