Download - Name Answer keys
Math 132 Exam #2 Fall 2018
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Signature _______________________________________________
Student ID Number ____ ____ ____ ____ ____ ____ ____ ____
Section Number _________________________________________
Section Instructor Day/Time Section Instructor Day/Time 1 Correia MWF 9:05 6 Farelli TuThu 1:00 2 Correia MWF 10:10 7 Farelli TuThu 2:30 3 Duan MWF 11:15 8 Destromp MWF 10:10 4 Nguyen MWF 12:20 9 Torres MWF 1:25 5 Hacking MWF 1:25 10 Dul MWF 8:00 11 Nakamura TuThu 8:30
• Calculators, papers, phones, smart watches, any device, or notes are not permitted on this exam.
The use of any of these items is considered Academic Dishonesty. • In the free response section, do not just give an answer. Clearly explain how you get it, providing
appropriate mathematical details. • An answer with no corresponding work will be awarded zero points. • This is a 2-hour exam.
Question Grade
Multiple Choice Questions (Out of 25)
6 (Out of 10)
7 (Out of 10)
8 (Out of 10)
9 (Out of 10)
10 (Out of 10)
11 (Out of 10)
12 (Out of 15)
Total (Out of 100)
Answer keys
2
Multiple Choice Section: Choose the one option that answers the question. There is no partial credit for questions 1-5. Only answers written in the answer blank will be graded. 1. [5 points] Which of the following is true about the series below?
∑𝑛
2𝑛2 + 4
∞
𝑛=1
(A) The series converges to 0 by the Test for Divergence.
(B) The series diverges by the Test for Divergence.
(C) The Test for Divergence is inconclusive for this series. Answer: 1. _________
(D) The series converges to 12 by the Test for Divergence.
____________________________________________________________________________________
2. [5 points] Which of the following is true about the series below?
∑ 9−𝑛∞
𝑛=1
(A) The series converges to 18
(B) The series converges to 98. Answer: 2. _________
(C) The series converges to 19.
(D) The series diverges.
:
3
3. [5 points] Suppose 𝑎𝑛 > 0 and 𝑏𝑛 > 0 for all 𝑛. If lim𝑛→∞
𝑎𝑛𝑏𝑛
= 1, which of the following is true?
(A) No conclusions about the convergence of ∑𝑎𝑛 can be made with the given information. (B) ∑𝑎𝑛 converges if ∑𝑏𝑛 converges. No conclusion about divergence of ∑𝑎𝑛 can be made. (C) ∑𝑎𝑛 diverges if ∑𝑏𝑛 diverges. No conclusion about convergence of ∑𝑎𝑛can be made. (D) ∑𝑎𝑛 converges if ∑𝑏𝑛 converges, and ∑𝑎𝑛 diverges if ∑𝑏𝑛 diverges.
Answer: 3. _________
____________________________________________________________________________________
4. [5 points] Consider the sequence below.
𝑎𝑛 = ln(3𝑛2 + 1) − ln (𝑛2 + 9) Which of the following is true? (A) The sequence converges to 0.
(B) The sequence converges to ln(3).
(C) The sequence converges to ln (2). Answer: 4. _________
(D) The sequence diverges.
D
B
4
5. [5 points] If 𝑎𝑛 = 2𝑛−1𝑛3+4
, which of the following is true?
I. ∑ 𝑎𝑛 ∞
𝑛=1
is convergent.
II. {𝑎𝑛}𝑛=1∞ is convergent.
III. ∑ 𝑎𝑛 ∞
𝑛=1
is divergent.
IV. {𝑎𝑛}𝑛=1∞ is divergent.
(A) I and II are true.
(B) III and IV are true.
(C) II and III are true. Answer: 5. _________
(D) I and IV are true.
A
5
Free Response Section: Show all work for each of the following questions. Partial credit may be awarded for questions 6-12. 6. [10 points] Determine if the integral converges or diverges. If it converges, find the value.
∫ 𝑥 ln(𝑥) 𝑑𝑥6
0
.
6
to Xhnxdx is convergentand
fbxfnxd ✗ =18hr(6) -9
0
6
7. [10 points] Determine if the series converges or diverges. Clearly state which test you used and show that this series meets the requirements to use this test.
∑4𝑛−1
𝜋𝑛 − 5
∞
𝑛=2
i.'
.
.
'
.
.
EI.IN?-sisdirergeutJby the comparisontest .
7
8. [10 points] Determine if the series converges or diverges. Clearly state which test you used and show that this series meets the requirements to use this test.
∑(−3)𝑛 ⋅ 𝑛4
(2𝑛)!
∞
𝑛=1
É Y÷; is convergentby the patio
TEST .
8
9. [10 points] Determine if the series converges or diverges. Clearly state which test you used and show that this series meets the requirements to use this test.
∑√𝑛2 + 4𝑛
𝑛3 + 5𝑛 + 1
∞
𝑛=1
É°F÷¥+,
is convergent
by thelimit comparison
TEST .
9
10. [10 points] Use the INTEGRAL TEST to determine if the series is convergent or divergent. Clearly show that this series meets the requirements to use this test.
∑4𝑛2
𝑛3 + 2
∞
𝑛=1
É,4n¥zisdirerg
10
11. [10 points] Determine if the series converges or diverges. Clearly state which test you used and show that this series meets the requirements to use this test.
∑𝑛𝑛
5𝑛+1 ⋅ (2𝑛 + 1)𝑛
∞
𝑛=1
É¥¥⇒ is
convergent bythe RootTEST
.
11
12. [15 points] Determine if the series is absolutely convergent, conditionally convergent, or divergent. Clearly state which test(s) you used and show that this series meets the requirements to use this test.
∑ (−1)𝑛 𝑛√7 + 𝑛3
∞
𝑛=1
-e
É,
I"?⇒tÉ¥is divergent by
the limit
comparison Est .
00
¥,
'Y¥n. converges bythe
Alternatingseries .
test .
Thus , ¥, c-Y-pyn.isconditionallyconvergent .