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Math 132, Spring 2011 - Exam 3
NAME:
STUDENT ID NUMBER:
This exam contains sixteen questions. The first fourteen are multiple choice questions and count for five points each. There is no partial credit on these questions, so read each question carefully, check your arithmetic and make sure that you have marked the answer you in- tended to mark. The last two questions, which are each worth fifteen points, require written answers, and some partial credit might be given. However, no credit will be given for information that is not germane to the problem at hand. Please make sure to write your name and stu- dent ID number on the pages that include your answers to the last two questions. In fact, you will get one point on each of these two questions for writing your name and ID number legibly.
- Let an = 3 โ (0.4)n^. The sequence {an}
(a) converges to โ 1 (b) converges to 0 (c) converges to 1 (d) converges to โ 3 (e) converges to 3 (f) diverges to โ (g) diverges to โโ (h) diverges but not to โ or โโ
- Which of the following sequences are monotonic?
(A)an = (^2) n 1 โ 1 (B)an = (โ1)
n n ,^ (C)an^ = 2n+3,^ (D)an^ =^ cos(nฯ)
(a) A only (b) A and B only (c) A and C only (d) A and D only (e) A, B and C only (f) A, B and D only (g) A, C and D only (h) A, B, C and D
- Compute the sum of the series โ^ โ n=
3(โ 2 1 )n
(a) 0 (b) 0. 5 (c) 1. 5 (d) โ 1. 5 (e) 2 (f) 6 (g) โโ (h) โ
- Determine whether the series โ^ โ n=
( (^) (n + 1)^12 โ (^) n^12 )
converges or diverges. If it converges, compute the sum.
(a) diverges to โ (b) diverges to โโ (c) diverges but not to โโ or โ (d) โ 1 (e) โ 2 / 3 (f) 0 (g) 1 (h) 4/ 5
- Which of the following is the best estimate of the error R in the approximation โ^ โ n=
n^3 โ^ s^100
(a) R โค 0. 001 (b) R โค 0. 005 (c) R โค 0. 0001 (d) R โค 0. 0005 (e) R โค 0. 00001 (f) R โค 0. 00005 (g) R โค 0. 000001 (h) R โค 0. 000005
- Apply the Ratio Test to โ^ โ n=
n! nn Find L and, if possible, decide whether the series converges or diverges. (a) L = 0 and the series converges by the Ratio Test. (b) L = 0 and the series diverges by the Ratio Test. (c) L = 1 and the Ratio Test fails. (d) L = 1 and the series converges by the Ratio Test. (e) L = eโ^1 and the series diverges by the Ratio Test. (f) L = eโ^1 and the series converges by the Ratio Test (g) L = โ and the series diverges by the Ratio Test. (h) L = โ and the series converges by the Ratio Test.
- Which of the following three alternating series is convergent? (A) โโ n=
(โ1)n^ n + 2^1 (B) โโ n=
(โ1)n^ n 4 n+ 9 (C) โโ n=
(โ1)ncos(ฯn)
(a) A only (b) B only (c) C only (d) A and B only (e) A and C only (f) B and C only (g) all (h) none
- Find the radius of convergence for the series โ^ โ n=
(2x)n n! (a) R = 0 (b) R = 1 (c) R = 1/ 2 (d) R = 2 (e) R = n! (f) R = x (g) R = โ (h) R cannot be determined
- For what values of x does the series โ^ โ n=
(3x + 1)n 10 n converge absolutely? (a) โ 1 < x < 1 (b) โ 1 < x < 3 (c) 0 < x < 3 (d) โ 3 < x < 3 (e) โ 11 / 3 < x < 3 (f) 0 < x < 10 (g) โ 1 / 10 < x < 10 (h) โ 8 < x < 12
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- Approximate the sum of the series โ^ โ n=
(โ1)n+1^10 nn
with an error less than 0.00005.
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- Use the Integral Test to decide whether โ^ โ n=
n (ln n)^2 converges or diverges. Show your work. Remember that you must show clearly that all conditions of the Integral Test are satisfied.
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