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MATH 152 Fall 2022 COMMON EXAM III VERSION B DIRECTIONS FIRST NAME print LAST NAME print INSTRUCTOR SECTION NUMBER 1 The use of a calculator laptop or computer is prohibited 2 TURN OFF cell phones and put them away If a cell phone is seen during the exam your exam will be collected and you will receive a zero 3 In Part 1 mark the correct choice on your ScanTron using a No 2 pencil The scantrons will not be returned therefore for your own records also record your choices on your exam 4 In Part 2 present your solutions in the space provided Show all your work neatly and concisely and clearly indicate your final answer You will be graded not merely on the final answer but also on the quality and correctness of the work leading up to it THE AGGIE CODE OF HONOR 5 Be sure to fill in your name UIN section number and version letter of the exam on the ScanTron form An Aggie does not lie cheat or steal or tolerate those who do Signature 1 This page left intentionally blank 2 PART I Multiple Choice 4 points each n 4 x 1 n3n n 1 1 Find the radius of convergence of the series1 a 4 b None of these c 3 d 4 e 0 2 Use a MacLaurin series to express f x xe2x 2 x a n n 2n2 x b n 2 x c 2n n 0 2n 4n 1 n 2n 1 n 0 n 0 2 as a power series centered at x 0 3 4 5 6 n 1 cnxn converges at x 4 and diverges at x 6 Which of the following statements is n 0 n 2n 1 2 x d n e None of these 3 Suppose that the series true cn4n converges I II cn7n diverges III n 1 cn5n may or may not converge a III only b I and II only c II and III only d I II and III e II only n 1 n 1 7 as a power series centered at 0 x3 n 0 4 Write f x 1 4x2 4 nx2n 3 a b n 4nx2n 3 4 nx2n c n 0 4 x d e n 4nx2n 6 0 2n 6 n 0 n 0 5 What is the value of the limit L that is used in the ratio test for this series a L b L 03 c L 2 d L 33 e L 4 6 Which of these is the MacLaurin series for f x x4 arctan 2x x 1 2 a n 2n 1 b n n 2n 1 2n 5 2n 5 0 0 n 8 n n 3 n 2n n 1 1 x 2n 5 n n 1 2 2n 1 x 1 2 c n 2n 1 d None of these 1 2 x e n 2n 1 2n 5 n 2n 0 0 9 n n n n 0 n 1 n 7 Find the radius of convergence of the series 4 x 3 n 1 a 4 b None of these c 4 d e 0 8 Find the 15th derivative at x 4 for f x 1 x 4 n3n a None of these b f 15 4 14 1 315 c f 15 4 15 315 d f 15 4 14 315 e f 15 4 1 15 315 1 9 If we find the Taylor Polynomial for f x a 10 76 b 6 7610 c 766 d 76 e 10 10 Suppose that 0 an bn for every positive integer n Which of the following statements is always true n 1 an is convergent then so is a If n 1 bn b If lim n bn 0 then n 1 an is convergent n 1 an is divergent then so is c If n 1 bn d none of these are always true an bn is divergent then so is e If centered at 7 what is the coefficient of the x 7 3 term n 1 n 1 x6 x3 10 n n5 n2 n 1 n 1 n 0 1 converges to s based on the Alternating Series Estimation Theorem which statement is 11 The series true 1 a R7 s s7 81 b R7 s s7 641 c R7 s s7 7 d None of these 1 e R7 s s7 49 12 Which series is conditionally convergent 7 a 1 b 1 c n 2n d n 1 e None of these 13 Which series is absolutely convergent 1 n a b None of these c 1 n 2n d 1 e n 1 n 0 n3 4 n n 0 n 1 n 1 n3 n n n n4 3 n 1 11 centered at n 0 a n 2n e 5 3 b c 1 5 32n 1 2n 14 Find the sum of the series 1 cos 5 3 3 5 1 sin 5 1 arctan 3 3 d None of these e 1 3 3 3 1 15 Find the Taylor polynomial T4 x the 4th degree Taylor polynomial for the function f x 1 5x2 a 0 a T4 x 1 5x2 25x4 125x6 b T4 x 1 5x2 25x4 125x6 c T4 x 1 5x2 25x4 125x6 d T4 x 1 5x2 25x4 e T4 x 1 5x2 25x4 16 Which of the following is the MacLaurin series for f x x sin x2 1 x a n 2n 1 1 x b n 2n 1 x c n 2n 1 d n 0 n 4n 2 n 4n 3 n 2n 2 n 4n 0 0 0 12 1 x 2n 1 1 x 2n n 4n 1 e n 0 13 PART II WORK OUT Directions Present your solutions in the space provided Show all your work neatly and concisely and Box your final answer You will be graded not merely on the final answer but also on the quality and correctness of the work leading up to it 17 5 points each Determine whether the following series converge or diverge Clearly explain your reasoning and state any tests used 6 5 cos n a n n 1 1 n n3 7 b n 1 14 18 8 points Find the MacLaurin series representation for the function f x 1 3x 2 1 15 19 10 points Find the radius of convergence and the interval of convergence of the power series You must test your endpoints for convergence 2 x 5 n8n n n n 1 16 20 8 points Find the Taylor series for f x xex about a 3 Express your answer in summation notation DO NOT WRITE IN THIS TABLE Question Points Awarded 1 16 17 18 19 20 Points 64 10 8 10 8100 17

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