Math 132
Exam 2
Fall 2014
Name:
ID Number:
Section Number:
Section
1
2
3
4
5
6
7
8
9
10
11
Instructor
Cook
Cook
Farelli
Farelli
Duanmu
McGibbon
McGibbon
Whitaker
Lowell
Nichols
Bates
Day/Time
MWF 9:05-9:55
MWF 10:10-11:00
MWF 11:15-12:05
MWF 12:20-1:10
MW 2:30-3:45
TuThu 8:30-9:45
TuThu 11:30-12:45
TuThu 2:30-3:45
MWF 1:25-2:15
MW 2:30-3:45
TuThu 8:30-9:45
• No papers or notes may be used.
• Please don’t just give an answer. Clearly explain how you
get it, providing appropriate mathematical details. An answer of ‘convergent’ or ‘divergent’ without work will
be awarded no points.
• This is a 2 hour exam.
Question
1
2
3
4
5
6
Total (out of 100)
Grade
1.
(a) [8 points] Determine whether the given improper integral converges or diverges. If it converges, what does
it converge to?
Z
e2
ln(t) dt
0
(b) [8 points] Does the sequence {an }, where
an =
n sin n
n2 + 1
converge or diverge? If it converges, find the limit.
2
2.
(a) [8 points] Consider the infinite series
75 375 1875
−
+
− ...
7
49
343
∞
X
Write this series in the form
an . Does this series
21 − 15 +
n=1
converge? If so, find the sum.
(b) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
nn
(−4)n
n=1
3
3.
(a) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
n=2
3
n(ln(n))2
(b) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
n=1
(−1)n
5n
4n + 7
4
4.
(a) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
n=1
6n
5n2 + 4n + 7
(b) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
n=1
7n
8n + 3
5
5.
(a) [8 points] Determine whether the series is convergent
or divergent. Clearly state which convergence test you
used.
∞
X
n=1
n!
20n · 62n+1
(b) [8 points] Using the Alternating Series Test, we can see
∞
X
(−1)n+1
converges. How many terms do we
that
2
n
n=1
need to estimate the sum with error less than 0.001?
6
6.
(a) [10 points] Determine whether the series is absolutely
convergent, conditionally convergent, or divergent. Clearly
state which convergence test(s) you used.
∞
X
n=1
2
nn
(−1) n
2
(b) [10 points] Determine whether the series is absolutely
convergent, conditionally convergent, or divergent. Clearly
state which convergence test(s) you used.
∞
2
X
n+1 n
(−1)
n3 − 3
n=2
7
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8