Math 132 Spring 2011 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 Let an 3 0 4 n The sequence an a converges to 1 b converges to 0 c converges to 1 d converges to 3 e converges to 3 f diverges to g diverges to h diverges but not to or 2 2 Let an 1 n1 n Compute limn an a 0 b 1 c e d 3 e f e3 g 3 h 3 3 Which of the following sequences are monotonic 1 A an 2n 1 1 n B an C an 2n 3 D an cos n n a A only b A and B only c A and C only d A and D only e A B and C only f A B and D only g A C and D only h A B C and D 4 4 Compute the sum of the series X 3 n 2 a 0 b 0 5 c 1 5 d 1 5 e 2 f 6 g h 5 1 n 2 5 Compute the sum of the series X n 1 2n 1 3 5n a 0 b c d 1 3 e 1 2 f 3 5 g 5 12 h 1 5 6 6 Determine whether the series X 1 1 2 n2 n 1 n 1 converges or diverges If it converges compute the sum a diverges to b diverges to c diverges but not to or d 1 e 2 3 f 0 g 1 h 4 5 7 7 Which of the following is the best estimate of the error R in the approximation X n 1 1 s100 n3 a R 0 001 b R 0 005 c R 0 0001 d R 0 0005 e R 0 00001 f R 0 00005 g R 0 000001 h R 0 000005 8 8 Which of the following three series is convergent X n 2 A 3 2 n 1 n 3n3 B 4 2 n 1 n 2n X 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 sin n n4 5 n 1 9 Apply the Ratio Test to X n n n 1 n Find L and if possible decide whether the series converges or diverges a L 0 and the series converges by the Ratio Test b L 0 and the series diverges by the Ratio Test c L 1 and the Ratio Test fails d L 1 and the series converges by the Ratio Test e L e 1 and the series diverges by the Ratio Test f L e 1 and the series converges by the Ratio Test g L and the series diverges by the Ratio Test h L and the series converges by the Ratio Test 10 10 Which of the following three alternating series is convergent X 1 A 1 n 2 n 1 n X n B 1 4 n 9 n 1 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 11 n C X 1 n cos n n 3 11 Which of the following three alternating series is absolutely convergent X 1 A 1 n 2 n 1 n X n B 1 4 n 9 n 1 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 n C X 1 n cos n n 3 12 Find the radius of convergence for the series 2x n n 1 n X a R 0 b R 1 c R 1 2 d R 2 e R n f R x g R h R cannot be determined 13 13 For what values of x does the series X n 1 3x 1 n 10n converge absolutely a 1 x 1 b 1 x 3 c 0 x 3 d 3 x 3 e 11 3 x 3 f 0 x 10 g 1 10 x 10 h 8 x 12 14 14 For which values of x can the function 1 f x 1 2x be expressed as a power series of the form X 1 n 2x n n 0 a all x b All x except x 1 2 c 1 x 1 d 1 2 x 1 2 e 2 x 2 f 9 x 9 g 29 x 29 h 92 x 92 15 Name Student ID 15 Approximate the sum of the series X 1 n 1 n 1 n 10n with an error less than 0 00005 16 Name Student ID 16 Use the Integral Test to decide whether X n 3 4 n ln n 2 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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