page 1 of 11 Math 1B Final Exam Friday 14 August 2009 Instructor Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Name Problem Number Score Maximum 1 ANSWERS 2 3 20 20 10 4 5 6 7 15 10 10 15 Total 100 Please do not begin this test until 2 10 p m You may work on the exam until 4 p m Please do not leave during the last 15 minutes of the exam time You must always justify your answers show your work show it neatly and when in doubt use words and pictures to explain your reasoning Please box your final answers Calculators are not allowed Please sign the following honor code I the student whose name and signature appear on this midterm have completed the exam by myself without any help during the exam from other people or from sources other than my allowed one page hand written cheat sheet Moreover I have not provided any aid to other students in the class during the exam I understand that cheating prevents me from learning and hurts other students by creating an atmosphere of distrust I consider myself to be an honorable person and I have not cheated on this exam in any way I promise to take an active part in seeing to it that others also do not cheat Signature Name ANSWERS Math 1B Final Exam 14 August 2009 page 2 of 11 1 20 pts total 2 pts each For each of the following statements determine if the conclusion ALWAYS follows from the assumptions if the conclusion is SOMETIMES true given the assumptions or if the conclusion is NEVER true given the assumptions You do not need to show any work or justify your answers for these question only your answer will be graded P a 2 pts If limn an 1 then 1 an converges ALWAYS SOMETIMES b 2 pts If the series P 1 an and P 1 NEVER bn both converge then SOMETIMES ALWAYS P 1 an bn converges NEVER c 2 pts If the sequences an and bn both diverge then an bn diverges ALWAYS d 2 pts If SOMETIMES PN 1 ALWAYS e 2 pts If ALWAYS NEVER an 10 for every N and if an 0 for every n then SOMETIMES P 1 cn 2n converges absolutely then SOMETIMES P 1 an converges NEVER P 1 cn 2 n converges conditionally NEVER Name ANSWERS f 2 pts If Math 1B Final Exam 14 August 2009 P 1 cn 2 n converges conditionally then ALWAYS P 1 cn converges absolutely SOMETIMES NEVER g 2 pts If bn is a decreasing positive sequence then ALWAYS page 3 of 11 P SOMETIMES 1 1 n bn converges NEVER h P 2 pts If p is a real number then the Ratio Test can be used to determine whether p 1 1 n converges ALWAYS i 2 pts If 0 an bn and ALWAYS SOMETIMES P 1 an converges then SOMETIMES j 2 pts If limn bn exists then ALWAYS P 1 NEVER P 1 bn converges NEVER bn bn 1 converges to b1 SOMETIMES NEVER Name ANSWERS Math 1B Final Exam 14 August 2009 page 4 of 11 2 20 pts total 5 pts each Determine whether each of the following series is ABSOLUTELY CONVERGENT CONDITIONALLY CONVERGENT or DIVERGENT You must specify which test s you use for each series and why the series satisfies the conditions of the test a 5 pts X 1 n n 1 n 1 n We use the ratio test We have lim n an 1 n 2 n 1 lim n an n 1 n n 2 0 1 lim n n 1 2 Therefore the series converges absolutely b 5 pts X 1 n n2 n 0 n 1 The series diverges by the divergence test limn n2 n 1 and so the limit of the alternating sequence is not 0 indeed it does not exist Name ANSWERS c 5 pts Math 1B Final Exam 14 August 2009 page 5 of 11 X 1 n n 2 ln n The series satisfies the conditions of the Alternating Series Test since ln n is an increasing positive function that tendsP to and so 1 ln n is descreasing positive and tends P to 0 But the absolute sequence 1 ln n diverges by for example comparison with 1 n 2 ln n n for every n and so 1 ln n 1 n Thus the series converges conditionally X 1 n d 5 pts n3 n n 1 P P 1 1 The series Thus n 1 n3 n converges by Comparison or Limit Comparison with n3 the above series converges absolutely Name ANSWERS Math 1B Final Exam 14 August 2009 page 6 of 11 3 10 pts Find the interval of convergence of the following power series X x 4 n n 2 n ln n 2n We begin with the ratio test to determine the radius of convergence We let cn 1 n ln n 2n Then ROC lim cn cn 1 n 1 ln n 1 2n 1 lim n n ln n 2n ln n 1 n 1 lim 2 lim n n n ln n 1 1 n 1 2 lim lim n 1 n 1 n n 2 lim 2 n n 1 if this limit exists n cancel and split the limits L Hospital evaluate and L Hospital The interval of convergence is centered at 4 and so is either 2 6 2 6 2 6 or 2 6 depending on the convergence at the endpoints We now test these endpoints At x 2 the series is X X 2 4 n 1 n n ln n 2n n ln n n 2 n 2 This series satisfies the conditions of the alternating series test as n ln n is the product of increasing positive functions and both tend to So the power series converges at x 2 At x 6 the series is X X 1 6 4 n n ln n 2n n ln n n 2 n 2 We use the integral test since 1 x ln x is a positive decreasing function We have Z Z 1 1 dx du ln u ln 2 x ln x u 2 ln 2 where we substituted u ln x Since the integral diverges the series does as well Thus the interval of convergence of the power series is x 2 6 Name ANSWERS Math 1B Final Exam 14 August 2009 page 7 of 11 4 15 pts Evaluate the following definite integral as a series Z 1 p 3 1 x6 dx 0 You must both Write your final answer in notation but you may leave your answer in terms of the binomial coefficients nk Write out the first four terms of the series We expand 3 1 x6 with the binomial theorem p X X 1 3 1 3 6n 3 6 n 6 1 x x x n n n 0 n 0 Z 1p Z 1X 1 3 6n 3 6 1 x dx x dx n 0 0 n …
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