Math 132 Exam 3 Fall 2016

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1 Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.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 clear will be part of the grade. Useful Formulas n i= i = n(n+) n i= i = n(n+)(n+) 6 ( ) n i= i3 = n(n+) sin θ + cos θ = + tan θ = sec θ + cot θ = csc θ sin(a ± B) = sin A cos B ± sin B cos A cos(a ± B) = cos A cos B sin A sin B tan(a ± B) = tan A±tan B tan A tan B sin A sin B = [cos(a B) cos(a + B)] cos A cos B = [cos(a B) + cos(a + B)] sin A cos B = [sin(a + B) + cos(a B)] sin x = ( cos x) cos x = ( + cos x) sin(θ) = sin θ cos θ cos(θ) = cos θ sin θ csc x dx = ln csc x + cot x + C sec x dx = ln sec x + tan x + C

2 Math 3 Exam 3 Page of. Determine whether the sequence defined by a n = ln(n 3 + ) ln(n 3 + n + 4) converges or diverges. If it converges, find the limit. A. 0 B. ln ( ) C. ln ( ) D. E. F. 9 + x n. Find ALL possible values of x for which the series converges. n A. It is not possible to find such x because the series diverges. B. x > 0 C. x < D. 3 < x < 3 E. 3 < x < F. < x < G. < x < n=0

3 Math 3 Exam 3 Page 3 of 3. Determine whether the sequence defined by ( a n = n cos n + π ) converges or diverges. If it converges, find the limit. A. B. C. 0 D. E. F. π 4. Which of the following sequences converge? a n = (n + )! (n + 4)!, b n = πn n 00, c n = ln(n0 ) n, d n = n 4 (n + )! A. {d n } only B. {a n }, {b n } only C. {c n }, {d n } only D. {a n }, {d n } only E. {a n }, {b n }, {d n } only F. {a n }, {c n }, {d n } only G. All of them

4 Math 3 Exam 3 Page 4 of. Assume the terms of a sequence {a n } are given by the following formula a n = 3n 3 + 3n n n 3n 3 Find the limit of the sequence or conclude that it diverges. A. 0 B. C. D. 3 E. 6 F Determine the value of the series n= 6 n(n + 3) or conclude that it diverges. A. 0 B. 3 C. 3 D. 3 6 E. 3 3 F. 0

5 Math 3 Exam 3 Page of 7. Determine the value of the series n + 3 n+ n=0 4 n or conclude that it diverges. A. 4 B. 9 C. 3 D E. F Assume a n is an infinite series with partial sums given by S N = 4 +. What is a N? n= A. B. C. 0 D. 3 0 E. - 9 F. - G. - 0 H

6 Math 3 Exam 3 Page 6 of 9. Which of the following series converge? I. n= ( ) n + n II. n= ln(n 4 ) III. n= n n + A. None of them B. I only C. II only D. III only E. I and II F. I and III G. II and III H. All of them 0. The series n= n = π Find the value of the series ( ) 4. n 8π 4 A. 4 8π 4 B. 4 ( ) π 4 C D. 8π 4 E. 90 8π 4 F n=

7 Math 3 Exam 3 Page 7 of. Which of the following series converge? I. n= n + n 4 4 n + n II. n= 4 n n + n III. n= n! 3 n A. None of them B. I only C. II only D. III only E. I and II F. I and III G. II and III H. All of them. Which of the following series converge? I. n= ln n e n + II. n= sin (n) n + n 3/ III. n= ( ) n n /3 A. None of them B. I only C. II only D. III only E. I and II F. I and III G. II and III H. All of them

8 Math 3 Exam 3 Page 8 of 3. Which of the following alternating series converge conditionally, but not absolutely? I. n= ( ) n n ln n II. n= ( ) n n(ln n) III. n= cos(πn) n 3 A. None of them B. I only C. II only D. III only E. I and II F. I and III G. II and III H. All of them 4. For which values of p does the series A. All values of p B. < p < C. p > D. p n= e n ( + e n ) p converge? E. p > F. p G. No values of p

9 Math 3 Exam 3 Page 9 of. Let a n be a series with partial sums S N. n= always true? Which of the following statements are I. If lim n a n = 0, then a n converges. n= II. If III. If n= n= a n = L, then lim n a n = L. a n converges, then lim n a n = 0. IV. If lim N S N = L, then A. None of them B. I and II C. I and III D. II and III E. III and IV F. I, II, IV G. II, III, IV H. All of them a n = L. n=

10 Math 3 Exam 3 Page 0 of Name: ID: Discussion Section Letter: You can find your discussion section on the front of your exam book Written Problem. You will be graded on the readability of your work. Use the back of this sheet, if necessary. 6. Determine whether the following series converge or diverge. State any tests used and show that all conditions of the test are satisfied. ln n (a) n n=3 (b) n= 3 n n n 3

11 Math 3 Exam 3 Page of Name: ID: Discussion Section Letter: You can find your discussion section on the front of your exam book Written Problem. You will be graded on the readability of your work. Use the back of this sheet, if necessary. 7. Determine whether the following series converge absolutely, converge conditionally, or diverge. State any tests used and show that all conditions of the test are satisfied. (a) ( ) n arctan(n) n= (b) n= ( ) n n(n + )(n 3) (c) n= ( ) n n n +

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