MATH 152 FALL SEMESTER 2008 COMMON EXAMINATION III VERSION A Name print Signature Instructor s name Section No INSTRUCTIONS 1 In Part 1 Problems 1 10 mark your responses on your ScanTron form using a No 2 pencil For your own record mark your choices on the exam as well 2 Calculators should not be used throughout the examination 3 In Part 2 Problems 11 15 present your solutions in the space provided Show all your work neatly and concisely and indicate your final answer clearly You will be graded not merely on the final answer but also on the quality and correctness of the work leading up to it 4 Be sure to write your name section number and version letter of the exam on the ScanTron form 1 Part 1 Multiple Choice 50 points Each question is worth 5 points Mark your responses on the ScanTron form and on the exam itself 1 Suppose that an and bn are sequences of real numbers Given that lim an 2 and n lim bn 3 determine lim a2n 3bn n n a 5 b 13 c 0 d 11 e 7 ln n n n 2 Compute lim a b 1 c 0 d 1 e 3 Determine the radius of the sphere given by the equation x2 y 2 2y z 2 1 0 a 1 b 2 c 3 d e 2 3 2 4 Suppose that 0 an bn for every positive integer n Which of the following statements is always true a If P b If n 1 P c If n 1 P an is convergent then so is an is divergent then so is d If lim bn 0 then n bn n 1 P bn is divergent then so is n 1 P an n 1 P bn n 1 P an is convergent n 1 e If lim an 0 then lim bn 0 n n 5 Consider the infinite series X 1 n n 1 n Which of the following statements is true n 1 a The Alternating Series Test shows that the series is convergent b The Test for Divergence shows that the series is divergent c The Ratio Test shows that the series diverges d The Ratio Test shows that the series converges e The series is absolutely convergent hence convergent 6 Suppose that bn is a sequence of positive numbers and that lim nbn 1 Which of the n bn following statements is true Hint nbn 1 n a The Limit Comparison Test shows that the series b The Limit Comparison Test shows that the series c The Ratio Test shows that the series d The Ratio Test shows that the series X X n 1 X bn is convergent bn is divergent n 1 bn is convergent n 1 X bn is divergent n 1 e The Test for Divergence shows that the series X n 1 3 bn is divergent 7 Compute the sum of the infinite series X 1 Hint Partial fractions n 2 n 3 n 1 a 1 3 b 1 c 2 d 1 4 e 1 5 8 Consider the following pair of infinite series X 1 n 1 I n3 4 n 1 II X 1 n 1 n4 3 n 1 Which of the following statements is true a Both series are absolutely convergent b I is absolutely convergent II is convergent but not absolutely convergent c II is absolutely convergent I is convergent but not absolutely convergent d I is convergent but not absolutely convergent II is convergent but not absolutely convergent e Both series are divergent 9 Compute the sum of the infinite series X 2n 1 3n n 1 a 2 b 4 c 2 3 d 1 e 6 10 Which of the following is the Maclaurin series expansion of the function f x cos x2 a X 1 n 2n x 2n n 0 b X 1 n x2n 1 2n 1 n 0 X 1 n 4n c x 2n n 0 X 1 n x4n 2 d 2n 1 n 0 e X 1 n 2n 2 x 2n n 0 4 Part 2 55 points Present your solutions to the following problems 11 15 in the space provided Show all your work neatly and concisely and indicate your final answer clearly You will be graded not merely on the final answer but also on the quality and correctness of the work leading up to it 11 Let T denote the triangle with vertices at A 1 1 1 B 2 3 3 and C 2 2 4 i 5 points Show that the sides AB and BC are perpendicular to each other ii 5 points Calculate the area of T 5 12 10 points Find the 3 rd degree Taylor polynomial of f x 2x4 x 1 at the point a 1 6 X 1 n 13 6 points Compute the sum of the infinite series n n 1 7 14 Consider the power series X x 1 n 2n n n 1 i 8 points Compute the radius of convergence of the power series ii 6 points Determine the interval of convergence of the power series Explain your reasoning concisely and completely 8 15 Let X 1 n 1 2n S x x 2n 1 n 1 x Z i 5 points Use the series given above to express S x dx as a power series Z ii 5 points Use the series obtained in i to express the definite integral S x dx as 0 the sum of an infinite series 9 1 2 iii 5 points Suppose that the definite integral in ii is approximated by the sum of the first 3 terms in the infinite series in ii Estimate the error of this approximation Justify your answer 10 QN PTS 1 10 11 12 13 14 15 TOTAL 11
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