Math 132 Exam 3 Fall 2016• 15 multiple choice questions worth 5 points each.• 2 hand graded questions worth 12 and 13 points each.• Exam covers sections 11.1-11.6:Sequences, Series, Integral, Comparison, Alternating, Absolute (not Root or Ratio)• No calculators!• For the multiple choice questions, mark your answer on the answer card.• Show all your work for the written problems. Your ability to make your solution clearwill be part of the grade.Useful FormulasPni=1i =n(n+1)2Pni=1i2=n(n+1)(2n+1)6Pni=1i3=n(n+1)22sin2θ + cos2θ = 11 + tan2θ = sec2θ 1 + cot2θ = csc2θsin(A ± B) = sin A cos B ± sin B cos A cos(A ± B) = cos A cos B ∓ sin A sin Btan(A ± B) =tan A±tan B1∓tan A tan Bsin A sin B =12[cos(A − B) − cos(A + B)]cos A cos B =12[cos(A − B) + cos(A + B)] sin A cos B =12[sin(A + B) + cos(A − B)]sin2x =12(1 − cos 2x) cos2x =12(1 + cos 2x)sin(2θ) = 2 sin θ cos θ cos(2θ) = cos2θ − sin2θRcsc x dx = −ln |csc x + cot x| + CRsec x dx = ln |sec x + tan x| + CMath 132 Exam 3 Page 2 of 111. Determine whether the sequence defined byan= ln(2n3+ 2) − ln(5n3+ 2n2+ 4)converges or diverges. If it converges, find the limit.A. 0B. ln25C. −ln25D.25E. 2F. −5G. Diverges2. Find ALL possible values of x for which the series∞Xn=09 + xn5nconverges.A. It is not possible to find such x because the series diverges.B. x > 0C. |x| < 1D. −3 < x < 3E. −3 < x < 5F. −5 < x < 5G. −∞ < x < ∞Math 132 Exam 3 Page 3 of 113. Determine whether the sequence defined byan= n2cos2n2+π2converges or diverges. If it converges, find the limit.A. −2B. −1C. 0D. 1E. 2F. πG. Diverges4. Which of the following sequences converge?an=(2n + 1)!(n + 4)!, bn=πnn100, cn=ln(n10)√n, dn=n4(n + 1)!A. {dn} onlyB. {an}, {bn} onlyC. {cn}, {dn} onlyD. {an}, {dn} onlyE. {an}, {bn}, {dn} onlyF. {an}, {cn}, {dn} onlyG. All of themMath 132 Exam 3 Page 4 of 115. Assume the terms of a sequence {an} are given by the following formulaan=13n3+223n3+323n3+ ··· +n23n3Find the limit of the sequence or conclude that it diverges.A. 0B. 1C.12D.23E.16F.19G. Diverges6. Determine the value of the series∞Xn=26n(n + 3)or conclude that it diverges.A. 0B.132C.53D.136E.133F.102G. DivergesMath 132 Exam 3 Page 5 of 117. Determine the value of the series∞Xn=02n−2+ 3n+14nor conclude that it diverges.A. 4B.92C.132D.374E.252F.978G. Diverges8. Assume∞Xn=1anis an infinite series with partial sums given by SN= 4 +2N. What is a5?A.225B.25C.110D.310E. -92F. -52G. -110H. -310Math 132 Exam 3 Page 6 of 119. Which of the following series converge?I.∞Xn=1(−1)n1 +1nII.∞Xn=21ln(n4)III.∞Xn=1n√n2+ 1A. None of themB. I onlyC. II onlyD. III onlyE. I and IIF. I and IIIG. II and IIIH. All of them10. The series∞Xn=11n4=π490. Find the value of the series∞Xn=22n4.A.8π445B.8π445− 1C. 16π490− 1D. −2E.8π490F.8π490− 16G. DivergesMath 132 Exam 3 Page 7 of 1111. Which of the following series converge?I.∞Xn=12n+ n44n+ n2II.∞Xn=14n5n+ nIII.∞Xn=1n!3nA. None of themB. I onlyC. II onlyD. III onlyE. I and IIF. I and IIIG. II and IIIH. All of them12. Which of the following series converge?I.∞Xn=1ln nen+ 2II.∞Xn=2sin2(n)n + n3/2III.∞Xn=1(−1)nn2/3A. None of themB. I onlyC. II onlyD. III onlyE. I and IIF. I and IIIG. II and IIIH. All of themMath 132 Exam 3 Page 8 of 1113. Which of the following alternating series converge conditionally, but not absolutely?I.∞Xn=2(−1)n√n ln nII.∞Xn=2(−1)nn(ln n)2III.∞Xn=1cos(πn)2n−3A. None of themB. I onlyC. II onlyD. III onlyE. I and IIF. I and IIIG. II and IIIH. All of them14. For which values of p does the series∞Xn=1en(2 + e2n)pconverge?A. All values of pB. −1 < p < 1C. p > 1D. p ≥ 1E. p >12F. p ≥12G. No values of pMath 132 Exam 3 Page 9 of 1115. Let∞Xn=1anbe a series with partial sums SN. Which of the following statements arealways true?I. If limn→∞an= 0, then∞Xn=1anconverges.II. If∞Xn=1an= L, then limn→∞an= L.III. If∞Xn=1anconverges, then limn→∞an= 0.IV. If limN→∞SN= L, then∞Xn=1an= L.A. None of themB. I and IIC. I and IIID. II and IIIE. III and IVF. I, II, IVG. II, III, IVH. All of themMath 132 Exam 3 Page 10 of 11Name:ID:Discussion Section Letter:You can find your discussion section on the front of your exam bookWritten Problem. You will be graded on the readability of your work.Use the back of this sheet, if necessary.16. Determine whether the following series converge or diverge. State any tests used andshow that all conditions of the test are satisfied.(a)∞Xn=3ln nn2(b)∞Xn=13n5n− n3Math 132 Exam 3 Page 11 of 11Name:ID:Discussion Section Letter:You can find your discussion section on the front of your exam bookWritten Problem. You will be graded on the readability of your work.Use the back of this sheet, if necessary.17. Determine whether the following series converge absolutely, converge conditionally, ordiverge. State any tests used and show that all conditions of the test are satisfied.(a)∞Xn=1(−1)narctan(n)(b)∞Xn=2(−1)nn(n + 1)(2n − 3)(c)∞Xn=2(−1)nn2n2+
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