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MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
Please note:
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SID:
TA:
Section(circle): Tuesday Thursday
Name:
1
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View Full Document MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
2
Problem 1.
Let
f
(
x
) =
∞
X
n
=0
3
n
2
n
+ 5
x
n
.
Find the value of
f
000
(0)
.
Solution.
If
c
n
denotes the coef±cient of
x
n
in this power series, then
f
000
(0) = 3!
·
c
3
= 3!
·
3
·
3
2
·
3+5
=
54
11
.
MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
3
Problem 2.
Determine if the series
∞
X
n
=0
1
(
n
+ 1)(
n
+ 2)
is convergent or divergent. If it is convergent, ±nd
its sum.
Solution.
We have that
1
(
n
+ 1)(
n
+ 2)
=
1
n
+ 1

1
n
+ 2
.
The series is telescoping:
1
1

1
2
+
1
2

1
3
+
1
3

1
4
+
···
+
1
n
+ 1

1
n
+ 2
= 1

1
n
+ 2
→
1
.
Hence the sum of the series is
1
.
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View Full Document MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
4
Problem 3.
Let
f
(
x
) = tan

1
x
. Find a power series representation for
f
around
0
. (
Hint:
represent
tan

1
x
as an integral).
Solution.
We have:
tan

1
x
=
Z
1
1 +
x
2
dx
+
C
=
Z
1
1

(

x
2
)
dx
+
C
=
Z
∞
X
n
=0
(

1)
n
x
2
n
+
C
=
∞
X
n
=0
(

1)
n
x
2
n
+1
2
n
+ 1
+
C.
To ±nd
C
, we evaluate both sides at
x
= 0
; since
tan

1
0 = 0
,
C
= 0
and
tan

1
x
=
∞
X
n
=0
(

1)
n
x
2
n
+1
2
n
+ 1
.
MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
5
Problem 4.
Is the improper integral
Z
∞
0
e
sin
x
dx
convergent or divergent? Explain.
Hint:
sin
x
is periodic.
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This note was uploaded on 02/22/2010 for the course MATH Math 31B taught by Professor Houdayer during the Spring '09 term at UCLA.
 Spring '09
 HOUDAYER

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