MAC 2312 - Fall 2023 - Practice Problems for Exam2
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Date
Apr 3, 2024
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MAC 2312 - Spring 2023
Practice Problems for EXAM 2
DISCLAIMER: This list of problems is not meant to be a preview of Exam 2, nor is it compre-
hensive of all possible variations for the problems in the topics included in Exam 2. It is meant
to provide you with a sense for the level of difficulty of the problems in Exam 2 and additional
practice for the free response problems. Several of these problems have been sourced from your
homework assignments and your textbook.
For the multiple choice portion of your exam, make sure to understand well the definitions used
in the exam topics, and to review the statements of the series tests covered in this test.
1. For each of the integrals below, determine whether it is improper, and whether it requires
partial fraction decomposition to be evaluated. Evaluate all integrals.
(a)
Z
2
x
2
−
9
x
−
9
x
3
−
9
x
dx
(b)
Z
4
1
x
(
x
2
−
9)
2
dx
(c)
Z
3
−∞
1
x
·
(1 + ln
|
x
|
)
dx
(d)
Z
20
x
(
x
−
1)
2
(
x
2
+ 1)
dx
(e)
Z
x
4
−
2
x
3
+
x
x
2
−
4
dx
(f)
Z
∞
−∞
arctan
x
x
2
+ 1
dx
(g)
Z
11
−
24
4
3
√
x
−
3
dx
(h)
Z
5
0
x
3
ln(3
x
)
dx
2. Determine whether the sequence converges or diverges. If it converges, find its limit.
(Note: assume
n
starts at 1 unless otherwise indicated.)
(a)
4
n
1 + 9
n
∞
n
=1
(b)
ln(2
n
2
+ 1)
−
ln(
n
2
−
3)
∞
n
=4
(c)
√
ne
−
n
∞
n
=1
(d)
(
−
1)
n
4
n
3
1 +
n
3
∞
n
=1
(e)
(
−
1)
n
4
n
2
1 +
n
3
∞
n
=1
(yes, it is not a typo)
(f)
a
n
=
e
8
n
(8
n
)
9
(g)
a
n
=
ln(2
n
)
n
2
(h)
a
n
= cos
2
n
+ 12
3. Is the sequence monotone? Bounded? Why / why not? Explain.
(Note: assume
n
starts at 1.)
(a)
√
ne
−
n
∞
n
=1
(b)
a
n
= 8
·
7
n
−
4
(c)
a
n
=
7
3
n
+ 2
(d)
a
n
=
8
9
n
(e)
a
n
=
−
n
+ 4
3
n
4. Determine whether the following geometric series are convergent. If they are, find their
sums.
(a)
∞
X
n
=1
2
n
5
n
−
1
(b)
∞
X
n
=0
7
n
+2
4
n
(c)
∞
X
n
=1
−
3
8
n
−
1
(d)
∞
X
n
=1
e
2
n
(
−
9)
1
−
n
5. Find the sum:
∞
X
n
=1
2
2
n
−
π
n
9
n
.
6. For the following telescoping series, find a formula for the
n
-th term of the sequence of
the partial sums (
s
n
).
Then evaluate
lim
n
→∞
s
n
to find the sum of the series or that the
series diverges.
(a)
∞
X
n
=1
1
2
n
−
1
2
n
+3
(b)
∞
X
n
=2
1
n
2
−
n
(c)
∞
X
n
=1
7
√
n
+ 5
−
7
√
n
+ 6
7. Where possible, use the Divergence Test to determine whether the following series are
divergent.
(a)
∞
X
n
=0
7
n
+2
4
n
(b)
∞
X
n
=1
2
n
5
n
−
1
(c)
∞
X
n
=1
1
−
2
n
3
n
−
2
(d)
∞
X
n
=4
(
ln(2
n
2
+ 1)
−
ln(
n
2
−
3)
)
(e)
∞
X
n
=1
√
n
·
e
−
n
8. Which of these series satisfy the hypotheses of the Integral Test? For those series, use
the Integral Test to determine whether they are convergent or divergent.
(a)
∞
X
n
=1
2
5
n
−
1
(b)
∞
X
n
=1
1
−
2
n
3
n
−
2
(c)
∞
X
n
=1
ln(
n
)
n
3
9. Determine whether each series is convergent or divergent using the Comparison or Limit
Comparison Test. Be sure to state which test you are using to support your conclusions
and show that the series satisfies the requirements of the test.
(a)
∞
X
n
=0
4
n
1 + 9
n
(b)
∞
X
n
=2
2
n
2
+ 1
n
7
−
3
n
4
+
n
(c)
∞
X
n
=1
4
n
−
1
n
3
(d)
∞
X
n
=0
cos
2
(
n
) + 2
7
n
+ 1
(e)
∞
X
n
=1
√
2
n
+ 3
n
(f)
∞
X
n
=1
7
n
√
16
n
4
−
2
10. Study each of the series below with each of the series tests seen so far (Divergence, Integral,
Comparison, or Limit Comparison). Does the series satisfies the requirements of the test?
Can the test be applied or not? Does the test determine whether the series is convergent
or divergent, or is it inconclusive? Fully justify your answers.
(a)
∞
X
n
=1
4
n
+ 3
2
n
(b)
∞
X
n
=1
4
n
4
1 +
n
3
(c)
∞
X
n
=1
1
3
√
3
n
+ 2
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