108 MthSc Spring 2006 PRINTED NAME:____________________
T4 V2
12.1 12.6, 12.8,12.9 Section #: ________
CUID____________________
Questions 1 12 are multiple choice questions. Enter your answer for each question on the omr sheet. You will NOT be penalized for incorrect answers. The omr sheet will not be returned so record your multiple choice responses on this test as well as the omr sheet. In order to receive full credit for questions 17 - 22, you must 1. Show legible and logical (relevant) justification which supports your final answer. If you are testing a series for convergence or divergence, you must name the test used, state the conditions for that test and how they are met, and interpret the results of the test. 2. Use correct and complete notation. 3. Present the answer in a SUMMARY sentence. If you are unaware of a questions meaning, please raise your hand and your instructor will try to clear up any confusion. NO CALCULATORS MAY BE USED ON THIS EXAM!
On my honor, I have neither given nor received inappropriate information during this exam.
SIGNATURE:__________________________________________________
Problem Number
17
18
19
20
21
22
Possible Points
10
8
10
8
8
8
Points Earned
MULTIPLE CHOICE__________ (out of 48) FREE RESPONSE: __________ (out of 52)
TOTAL __________
1
MthSc 108 Spring 2006
T4 V2
12.1 12.6, 12.8,12.9
Multiple Choice Questions. Choose the ONE correct answer for each question. Mark your answer on the scantron sheet. (48 pts)
1.
If
a
n =1
n
= 1 , then lim an =
n
a. 4 b. 0 c. 1
d. 2 e. cannot be determined
2. Find the radius of convergence of the series a. R = 0
n3 x n . n n =1 3
b. R = 1 c. R=
1 3
d. R = 3 e. R =
3. Identify the ONE correct statement. a.
{an } = {an } = {an } = {an } = {an } =
1 diverges; n 1 diverges; n 1 converges; n 1 diverges; n 1 converges; n
{an } = (1)n
1 converges; n 1 diverges; n
n diverges; (1)
n =1 n =1
1
n
1 converges n 1 converges n
n
b.
{an } = (1)n
n converges; (1)
n =1 n =1
1
n
c.
{an } = (1)n
1 converges; n
n diverges; (1)
n =1 n =1
1
1 converges n
d.
{an } = (1)n
1 converges; n 1 converges; n
1 n diverges; n =1
(1)
n =1
n
1 diverges n
n
e.
{an } = (1)n
n converges; (1)
n =1 n =1
1
1 diverges n
2
MthSc 108 Spring 2006
T4 V2
12.1 12.6, 12.8,12.9
4. Find a formula for the general term an , n 1, of the sequence, assuming that the pattern of the first few terms continues. 27 3 9 ,, . . . 1, - , 7 49 343
a. b. c. d. e.
4 7
n1
3 ( 1) 7
n1
3 ( 1)n 1 7 3 ( 1)n 7 4 ( 1)n 7
n 1
n 1
n 1
5. Find the radius of convergence of the series a. b. c. d. e.
(3 x 2)n . n 3n n =1
R =1 1 3 5 R= 3 R=2 R=
R=0
6. The sum of the geometric series 2 1 +
1 1 1 1 + + . . . is 2 4 8 16
a. b. c.
4 3 5 4
1
d. 4 e.
2 3
3
MthSc 108 Spring 2006
T4 V2
12.1 12.6, 12.8,12.9
7. The sequence 3r n +1 converges if and only if a. b. c. d. e.
{
}
0< <1
r r >1 1 < r 1 r 1 r <1
8. The series a. b. c. d. e.
5
n 4 + 15 diverges by comparison with the series bn . Find 5 n n =1 n =1
b
n =1
n
.
b = n
n =1 n =1 n =1 n =1 n =1 n n =1 n =1 n =1 n =1 n =1 n
b = n b = n
n
1
5
1 1
4
b = n
n
b = n
n
9. Find all values of x for which the series a. b. c.
d. e. (0, 2) [0, 2] [ 1, 1 ] [0, 2) [ 1, 1)
( x 1)n n converges. n =1
10. If the radius of convergence of the power series series
c x
n=0 n
n
is R = 4, the radius of convergence of the power
nc x
n =1 n
n 1
a. equals
b. cannot be determined from the information given c. equals 0
d. equals 1 e. equals 4
4
MthSc 108 Spring 2006
n
T4 V2
12.1 12.6, 12.8,12.9
11. sn = ai = a1 + a2 + a3 + . . . + an is the nth partial sum of a series defined as
i =1
a
n =1
n
The following statements concern the relationship of the sequence of partial sums {sn } and the series
a
n =1
n
. Select the ONE false statement from the choices given.
a. If lim an = 0 , then
x
a
n =1
n
must be convergent.
b. If c. If
a
n =1
n
converges, then {sn } must also be convergent. converges, then lim an = 0
n
a
n =1
n
d. If {sn } converges, then
a
n =1
n
must also be convergent.
12. Complete the statement. The series
3 6
n n =1
1 n
a. converges and the sum is 6. b. diverges. c. converges and the sum is
1 . 2
d. converges and the sum is 3. e. converges, but the sum cannot be determined.
For All Written Problems: State the test that you are using (except in #13 when you are using the Integral Test) State the necessary conditions for the test that you are using and explain how they are met. Give your conclusion in a sentence.
13. Use the Integral Test to determine whether the series
n=3
n ln n ...