Math 132 Exam 2 Spring 2015 Name ID Number Section Number Section Instructor Section Instructor Farelli Farelli Clark Clark Brown Brown Duanmu Oloo Day Time MWF 10 10 MWF 9 05 MWF 11 15 MWF 12 20 MW 2 30 MW 4 00 TuThu 8 30 TuThu 10 00 Day Time Benincasa TuThu 1 00 Benincasa TuThu 2 30 MWF 10 10 MWF 12 20 MWF 1 25 Buckman TuThu 11 30 TuThu 1 00 TuThu 2 30 Buskin Yaping Yaping Wen Wen 9 10 11 12 13 15 16 17 1 2 3 4 5 6 7 8 No papers or notes may be used Please don t just give an answer Clearly explain how you get it pro viding appropriate mathematical details An answer of convergent or divergent with no supporting work will be awarded zero points This is a 2 hour exam Question MC Total 6 7 8 9 Total out of 100 Grade cid 107 cid 107 cid 107 cid 107 1 Mutiple Choice Section Choose the one option that best answers the question There is no partial credit for questions 1 5 1 5 points Consider the following an n3 3n3 5 I an converges II an diverges III an converges IV an diverges cid 80 n 1 cid 80 n 1 Which of the following is true 2 5 points Which of the following are true for the Test for Divergence an does not exist then an diverges an exists but is not 0 then an diverges an 0 then an converges cid 80 n 0 cid 80 n 0 cid 80 n 0 a I III b I IV c II III d II IV I If lim n II If lim n III If lim n a I II b I III c I II III d II III 2 3 5 points Consider two positive series Which of the following is de nitely true cid 88 cid 88 an and bn with an bn a If an converges then bn converges b If bn diverges then an diverges cid 88 cid 88 cid 88 cid 88 cid 88 cid 88 c If bn converges then d If lim n verge an bn an converges cid 88 an and cid 88 0 then both bn converge or both di 4 5 points Determine if the following series converges and if so the value it converges to cid 88 n 1 1 n 7 4n 3n 5 5 points Which of the following yields an inconclusive result for the a 21 b 4 c 3 d divergent Root Test a lim n b lim n c lim n d lim n n cid 112 an 1 n cid 112 an n cid 112 an L 1 n cid 112 an L 1 Please ll in your letter answer for questions 1 5 below 1 2 3 4 5 3 Free Response Portion Show all work for each of the following questions Partial credit may be awarded for questions 6 9 You will receive no credit for an answer of convergent or divergent without supporting work 6 a 5 points Evaluate the following integral cid 90 y y2 2y 3 dy b 10 points Does the following integral converge or diverge If it converges what does it converge to cid 90 3 2 1 3 x dx 4 7 a 10 points Find the values of x for which the series is convergent cid 88 n 0 3 n 1 x 4 n b 10 points Does the series converge or diverge State which test you used and clearly show that the series meets the conditions to use this test cid 88 n 1 cos2 n 5n 5 8 a 10 points Does the series converge or diverge State which test you used and clearly show that the series meets the conditions to use this test cid 88 n 1 4n 6n2 2n 8 b 10 points Does the series converge or diverge State which test you used and clearly show that the series meets the conditions to use this test cid 88 n 3 ln n2 n 6 9 a 10 points Does the series absolutely converge conditionally con verge or diverge State which test you used and clearly show that the series meets the conditions to use this test cid 88 n 1 n3 n 1 enn 7 b 10 points Does the series absolutely converge conditionally con verge or diverge State which test you used and clearly show that the series meets the conditions to use this test cid 88 n 2 1 n 1 5 n 3 3 8 This page is intentionally left blank for additional work 9