Math 2224 Common Exam Spring 2002
FORM A
Instructions:
Please enter your NAME, your ID NUMBER, the FORM DESIGNATION
LETTER and your INDEX NUMBER on the op-scan sheet. The index number should
be written in the upper right-hand box labeled "Course". Darken the appropriate circles
below the ID number and form designation letter.
Use a No.2 pencil; machine grading
may ignore faintly marked circles.
Mark your answers to the test questions in rows 1 - 15 of the op-scan sheet. Your score on
this part of the test will be the number of correct answers. You have one hour to complete
this part of the nal exam.
[
1
] Let
S
n
=
n
X
k
=1
a
k
:
If
S
n
=
2
n
2
+3
n
2
+1
for each
n
1
;
which of the following statements is
correct ?
1)
1
X
k
=1
a
k
is divergent 2)
1
X
k
=1
a
k
is convergent and its sum is 3
:
3) lim
k
!1
a
k
=2 4) lim
k
!1
a
k
=0
[
2
] Consider the series
1
X
n
=1
sin(
1
n
)
:
1) it diverges by the limit comparison test 2) it converges by the limit comparison test
3) it converges, because lim
n
!1
sin(
1
n
)=0
:
4) it converges by the ratio test
5) it diverges by the ratio test
[
3
] What is the open interval of convergence of the power series
1
X
n
=1
(4
x
1)
n
n
2
n
?
1) (0
;
1
2
) 2) (
1
4
;
3
4
) 3) (
1
;
1) 4) It converges only at
x
=
1
4
:
[
4
] Find the sum of the series
1
X
n
=3
2
n
2
3
n
1
=
2
9
+
4
27
+
8
81
+
:
1)
2
3
2)
2
9
3)
1
3
4)
2
5
[
5
] Find a power series representation for the function
f
(
x
)=
1
(1
x
)
2
1)
1
X
n
=1
x
n
1
2)
1
X
n
=1
nx
n
1
3)
1
X
n
=1
(
1)
n
1
x
n
1
4)
1
X
n
=1
(
1)
n
1
nx
n
1
1
