page 1 of 11Math 1B: Final ExamFriday, 14 August 2009Instructor: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Name:Problem Number 1 2 3 4 5 6 7 TotalScoreMaximum 20 20 10 15 10 10 15 100Please 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 examby myself, without any help during the exam from other people, or from sources other thanmy allowed one-page hand-written cheat sheet. Moreover, I have not provided any aid toother students in the class during the exam. I understand that cheating prevents me fromlearning and hurts other students by creating an atmosphere of distrust. I consider myselfto be an honorable person, and I have not cheated on this exam in any way. I promise totake an active part in seeing to it that others also do not cheat.Signature:Name: Math 1B Final Exam: 14 August 2009 page 2 of 111. (20 pts total – 2 pts each) For each of the following statements, determine if the conclusionALWAYS follows from the assumptions, if the conclusion is SOMETIMES true given theassumptions, or if the conclusion is NEVER true given the assumptions. You do not need toshow any work or justify your answers for these question: only your answer will be graded.(a) (2 pts) If limn→∞an< 1, thenP∞1anconverges.ALWAYS SOMETIMES NEVER(b) (2 pts) If the seriesP∞1anandP∞1bnboth converge, thenP∞1(an+ bn) converges.ALWAYS SOMETIMES NEVER(c) (2 pts) If the sequences {an} and {bn} both diverge, then {anbn} diverges.ALWAYS SOMETIMES NEVER(d) (2 pts) IfPN1an≥ −10 for every N and if an≤ 0 for every n, thenP∞1anconverges.ALWAYS SOMETIMES NEVER(e) (2 pts) IfP∞1cn2nconverges absolutely, thenP∞1cn(−2)nconverges conditionally.ALWAYS SOMETIMES NEVERName: Math 1B Final Exam: 14 August 2009 page 3 of 11(f) (2 pts) IfP∞1cn(−2)nconverges conditionally, thenP∞1cnconverges absolutely.ALWAYS SOMETIMES NEVER(g) (2 pts) If {bn} is a decreasing positive sequence, thenP∞1(−1)nbnconverges.ALWAYS SOMETIMES NEVER(h) (2 pts) If p is a real number, then the Ratio Test can be used to determine whetherP∞11/npconverges.ALWAYS SOMETIMES NEVER(i) (2 pts) If 0 ≤ an≤ bnandP∞1anconverges, thenP∞1bnconverges.ALWAYS SOMETIMES NEVER(j) (2 pts) If limn→∞bnexists, thenP∞1(bn− bn+1) converges to b1.ALWAYS SOMETIMES NEVERName: Math 1B Final Exam: 14 August 2009 page 4 of 112. (20 pts total – 5 pts each) Determine whether each of the following series is ABSOLUTELYCONVERGENT, CONDITIONALLY CONVERGENT, or DIVERGENT. You must specifywhich test(s) you use for each series, and why the series satisfies the conditions of the test.(a) (5 pts)∞Xn=1(−1)n(n + 1)n!(b) (5 pts)∞Xn=0(−1)nn2n + 1Name: Math 1B Final Exam: 14 August 2009 page 5 of 11(c) (5 pts)∞Xn=2(−1)nln n(d) (5 pts)∞Xn=1(−1)nn3+√nName: Math 1B Final Exam: 14 August 2009 page 6 of 113. (10 pts) Find the interval of convergence of the following power series:∞Xn=2(x − 4)nn ln n 2nName: Math 1B Final Exam: 14 August 2009 page 7 of 114. (15 pts) Evaluate the following definite integral as a series.Z103p1 + x6dxYou must both:• Write your final answer in Σ notation, but you may leave your answer in terms of thebinomial coefficientskn.• Write out the first four terms of the series.Name: Math 1B Final Exam: 14 August 2009 page 8 of 115. (a) (5 pts) Find power series representations (centered at 0) for each of the following func-tions, and state the intervals of convergence for each series• ln(1 − x)• ln(1 − x2)• ln(1 + x)(b) (5 pts) Show that as power series, the representations for the above functions satisfyln(1 − x2) = ln(1 − x) + ln(1 + x)Name: Math 1B Final Exam: 14 August 2009 page 9 of 116. (a) (2 pts) State the power series representation for arctan(x) centered at 0. What is itsinterval of convergence?(b) (4 pts) What is tan π/6? Use this value for x in the power series representation to finda series that converges to π/6. Is the convergence absolute or conditional?(c) (4 pts) Use the error estimate for an Alternating Series to determine how many sum-mands you would need in order to use this series to estimate π/6 correct to three decimalplaces (|error| < 0.001).Name: Math 1B Final Exam: 14 August 2009 page 10 of 117. (15 pts) Find a power-series representation (centered at 0) for the solution to the followinginitial value problem:y00− xy0− 2y = 0, y(0) = 1, y0(0) = 1Name: Math 1B Final Exam: 14 August 2009 page 11 of 11References: All the problems on this midterm are due to the instructor, although they areloosely based on the material in Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart. The honor-code language is adapted from the Stanford Honor Code (http://www.stanford.edu/dept/vpsa/judicialaffairs/guiding/honorcode.htm) and from the examsby Zvezda Stankova.Feel free to use this page for extra scrap
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