MATH 141 EXAMINATION II Name November 10 2004 ID Section This examination will be machine processed by the University Testing Service Use only a number 2 pencil on your scantron On your scantron identify your name this course Math 141 and the date Code and blacken the corresponding circles on your scantron for your student I D number and class section number Code in your test form There are 13 multiple choice questions worth a total of 65 points For the problems 1 to 13 either four or five possible answers are given only one of which is correct You should solve the problem circle the letter of your answer in the exam form and blacken the corresponding space on the scantron Mark only one choice darken the circle completely There are 7 short answer and 1 partial credit questions 35 points Each of the 7 multiple choice questions 1 5 8 9 10 11 12 are designed so that partial credit may be given since your answer indicates whether you got part of the question correct In order to obtain full credit for the partial credit problem all work must be shown Credit will not be given for an answer not supported by work THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION There are 18 problems on 11 pages including this one Check your booklet now The area below is for the instructor s use 17 12 18 11 Total 23 MATH 141 EXAMINATION II PAGE 2 1 5 pts Determine which of the following expressions are indeterminate forms DO NOT compute their limits 1 i lim 1 tan x x x 0 ii lim x 0 sin 2x ln x a Only i and ii are indeterminate b Only i is indeterminate c Only iii is indeterminate d Only i and iii are indeterminate e Only ii and iii are indeterminate tan x x 0 ln 1 x 2 5 pts Compute the value of lim a 0 b 1 c d e iii lim x ln x x MATH 141 3 5 pts Consider the series EXAMINATION II X n 2 n2 2 1 a The series diverges b The series converges to 11 6 c The series converges to 0 d The series converges to 3 2 e None of the above 4 5 pts Consider the series X xn n 1 2n a The series converges if x 2 to 2 2 x 2 x 2 x c The series converges if x 2 to 2 x b The series converges if x 2 to d The series converges if x 2 to e The series diverges for all x 2 x x PAGE 3 MATH 141 EXAMINATION II PAGE 4 5 5 pts Which of the following statements are true X X 1 1 i diverges by comparison with n n 4 n 1 n 1 X X 1 1 ii diverges by limit comparison with n n 4 n 1 n 1 a Only i is true b Only ii is true c Both are true d Neither is true X 1 convergent Hint Use the n ln n p n 2 integral test and the substitution u ln x to integrate 6 5 pts For what values of p is the infinite series a p 1 b p 1 c p 1 d 1 p 1 e For all p MATH 141 EXAMINATION II PAGE 5 X 1 7 5 pts Use an improper integral to find the smallest number of terms of the series n3 n 1 required to ensure that the the bound obtained for the error in the sum is less than 0 005 a 110 b 100 c 50 d 11 e 5 8 5 pts Which of the following two series converge i X n 1 2 n n a Only i converges b Only ii converges c Both converge d Neither converge ii X 1 en n 1 2n MATH 141 EXAMINATION II PAGE 6 9 5 pts Which of the following two series converge i X 2n n 1 n ii X n 2 n3 1 4 a Only i converges b Only ii converges c Both converge d Neither converge X cos n first write out the first few terms in the series and n then determine which of the following statements are true 10 5 pts Given the series n 1 i The series diverges by the test for divergence X 1 ii The given series diverges by comparison with n n 1 iii The given series converges by the alternating series test a None are true b Only i is true c Only ii is true d Only iii is true e Only i and ii are true MATH 141 EXAMINATION II PAGE 7 11 5 pts Which of the following statements are true n X ln n i The series diverges by the test for divergence n n 2 n X ln n converges by the root test ii The series n n 2 n X 2 converges by the root test iii The series 1 n n 1 a Only ii is true b None are true c Only i is true d Only ii and iii are true e Only iii is true 12 5 pts Suppose X an and n 1 X bn are two series with positive terms and n 1 n 1 gent Which of the following statements are always true X i 1 n 1 an is convergent n 1 ii If bn an for all positive integers n then iii lim cos an 1 n a Only i is true b Only ii is true c Only i and ii are true d Only i and iii are true e All three are ture X X n 1 bn is also convergent an is conver MATH 141 EXAMINATION II PAGE 8 13 5 pts Which of the following statements would allow you to conclude that the series X n 1 is divergent a lim n b an p n an 1 1 n2 for all n c lim an 1 n d lim n an 1 1 an 1 e an n for all n an MATH 141 EXAMINATION II PAGE 9 For each series below determine whether it is absolutely convergent conditionally convergent or divergent Please code on your answer sheet A if the series is Absolutely convergent C if the it is Conditionally convergent or D if it is Divergent X 1 n 1 14 4 pts 3 n n 1 15 4 pts X 2 n 7 n 1 16 4 pts X n 2 3n 1 n ln n ln n3 MATH 141 EXAMINATION II PAGE 10 17 12 pts 3 pts each For each of the following determine if the sequence an n 1 converges or diverges If it converges find the limit of the sequence Please circle your answers a an n 1 n 1 n2 Divergent or Converges to b an 1 n 1 n n3 1 Divergent or Converges to c an n n 1 Divergent or Converges to d an e 1 n …