Math 132 Spring 2009 Exam 3 NAME STUDENT ID NUMBER This exam contains sixteen questions The first fourteen are multiple choice questions and count for five points each There is no partial credit on these questions so read each question carefully check your arithmetic and make sure that you have marked the answer you intended to mark The last two questions which are each worth fifteen points require written answers and some partial credit might be given However no credit will be given for information that is not germane to the problem at hand Please make sure to write your name and student ID number on the pages that include your answers to the last two questions In fact you will get one point on each of these two questions for writing your name and ID number legibly 1 1 Suppose dy 1 y dx and y 0 1 Find y 4 a 0 b 2 c 3 d 4 e e4 f ln 4 g 1 4 h 1 4 2 2 Solve the differential equation dy 2 x ex y dx with initial condition y 0 0 2 a ey ex 1 2 b ey 12 ex 1 2 2 c ey 12 ex 2 d ey ex 2 e e y ex 1 2 f e y 12 ex 1 2 2 g e y 12 ex 2 h e y ex 3 3 Let an 1 n n The sequence an a converges to 1 b converges to 0 c converges to 1 d converges because it is an alternating series e diverges because it is an alternating series f diverges to g diverges to h diverges but not to or 4 n 4 Let an n 3 n Compute limn an a 0 b 1 c e d 3 e f e3 g 3 h 5 5 Compute the sum of the series X n 2 a 0 b c 7 12 d 28 3 e 7 3 f 21 4 g 21 16 h 7 6 7 4n 6 Compute the sum of the series X n 1 3n 1 1 6n 1 a 0 b c d 1 3 e 1 2 f 1 6 g 1 3 h 4 5 7 7 Compute the sum of the series X n 1 1 1 cos cos n n 1 a diverges to b diverges to c diverges but not to or d 1 e 1 cos 1 f 0 g 1 h cos 1 8 8 Which of the following three series is convergent A X n 2 3 2 n 1 n B X 3 4 2 n 1 n 2n a A only b B only c C only d A and B only e A and C only f B and C only g all h none 9 C X n 1 n n4 5 9 Apply the Ratio Test to 2n n 1 n X Find and if possible decide whether the series converges or diverges a 0 and the series converges by the Ratio Test b 0 and the series diverges by the Ratio Test c 0 and the Ratio Test fails d 2 and the series converges by the Ratio Test e 2 and the series diverges by the Ratio Test f 2 and the Ratio Test fails g and the series diverges by the Ratio Test h and the Ratio Test fails 10 10 Apply the Root Test to X 1 2 n n n 1 Find and if possible decide whether the series converges or diverges a e 2 and the series converges by the Root Test b e 2 and the series diverges by the Root Test c e 2 and the Root Test fails d 1 and the series converges by the Root Test e 1 and the series diverges by the Root Test f 1 and the Root Test fails g and the series diverges by the Root Test h and the Root Test fails 11 11 Which of the following three alternating series is convergent A X 1 n n 1 n 1 n 2 B a A only b B only c C only d A and B only e A and C only f B and C only g all h none 12 X 1 1 n n n 1 C X 1 n n 1 ln n n 12 Which of the following three alternating series is absolutely convergent A X 1 n n 1 n 1 n 2 B a A only b B only c C only d A and B only e A and C only f B and C only g all h none 13 X 1 1 n n n 1 C X 1 n n 1 ln n n 13 Find the radius of convergence for the series 3nxn n 1 n 2 X a R 0 b R 1 c R 1 2 d R 3 e R 3 2 f R 1 g R x h R 14 14 For what values of x does the series X n 1 x 2 n 10n converge absolutely a 1 x 1 b 1 x 3 c 0 x 3 d 0 x 10 e 8 x 12 f 2 x 12 g 12 x 12 h 10 x 10 15 Name Student ID 15 The series 1 1 t t2 t3 t n 1 t converges on the open interval 1 t 1 Find the power series for ln 1 x and compute its radius of convergence Show your work 16 Name Student ID 16 Use the Integral Test to decide whether X 4 n 2 n ln n converges or diverges Show your work Remember that you must show clearly that all conditions of the Integral Test are satisfied 17 Name Student ID 18 Name Student ID 19
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