Math 132 Exam #2 Fall 2018

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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 .