2022W2_MATH_101C_ALL_2022W2_Integral_Cal.56O9DSPQEY00.WW09
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University of British Columbia *
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Course
101
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
10
Uploaded by DeanComputer3393
Daniel Truong
2022W2
MATH
101C
ALL
2022W2
Integral
Cal
Assignment WW09 due 03/30/2023 at 11:59pm PDT
Problem 1.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Classify each of the following statements as either true or false.
a)
If
a
n
≤
b
n
for all
n
and the series
∞
∑
n
=
1
b
n
converges, then the
series
∞
∑
n
=
1
a
n
converges.
?
b) If
S
n
=
n
∑
k
=
1
a
k
and lim
n
→
∞
S
n
+
1
S
n
=
1
3
, then
∞
∑
n
=
1
a
n
converges.
?
c) If
a
n
≥
0 for all
n
and the series
∞
∑
n
=
1
a
n
converges, then
∞
∑
n
=
1
(
a
n
)
2
converges.
?
d)
If lim
n
→
∞
S
n
=
0 where
S
n
=
n
∑
k
=
1
a
k
, then the series
∞
∑
k
=
1
a
k
con-
verges.
?
e) If lim
n
→
∞
a
n
=
0, then the series
∞
∑
n
=
1
a
n
converges.
?
f)
If the series
∞
∑
n
=
1
a
n
converges, then lim
n
→
∞
S
n
=
0 where
S
n
=
n
∑
k
=
1
a
k
.
?
Answer(s) submitted:
•
False
•
True
•
True
•
True
•
False
•
False
submitted: (correct)
recorded: (correct)
1
Problem 2.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Analyze each of the following attempted proofs, and decide if the
given argument is correct or incorrect.
Each of the given statements appeals to the Comparison Test (and
NOT the Limit Comparison Test.)
Note
: If the conclusion is true but the justification provided is
flawed, the appropriate answer is ”Incorrect”.
a) For all
n
>
2,
1
n
2
-
6
<
1
n
2
, and the series
∞
∑
n
=
1
1
n
2
converges, so
by the Comparison Test, the series
∞
∑
n
=
1
1
n
2
-
6
converges.
?
b) For all
n
>
2,
log
(
n
)
n
2
>
1
n
2
, and the series
∞
∑
n
=
1
1
n
2
converges,
so by the Comparison Test, the series
∞
∑
n
=
2
log
(
n
)
n
2
converges.
?
c) For all
n
>
1,
n
4
-
n
3
<
1
n
2
, and the series
∞
∑
n
=
1
1
n
2
converges, so
by the Comparison Test, the series
∞
∑
n
=
1
n
4
-
n
3
converges.
?
d) For all
n
>
1,
1
n
log
(
n
)
<
2
n
, and the series 2
∞
∑
n
=
1
1
n
diverges, so
by the Comparison Test, the series
∞
∑
n
=
2
1
n
log
(
n
)
diverges.
?
e) For all
n
>
2,
n
n
3
-
7
<
2
n
2
, and the series 2
∞
∑
n
=
1
1
n
2
converges,
so by the Comparison Test, the series
∞
∑
n
=
1
n
n
3
-
7
converges.
?
f) For all
n
>
1,
arctan
(
n
)
n
3
<
π
2
n
3
, and the series
π
2
∞
∑
n
=
1
1
n
3
con-
verges, so by the Comparison Test, the series
∞
∑
n
=
1
arctan
(
n
)
n
3
con-
verges.
?
Answer(s) submitted:
•
Incorrect
•
Incorrect
•
Incorrect
•
Incorrect
•
Correct
•
Correct
submitted: (correct)
recorded: (correct)
2
Problem 3.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
For each of the following series decide if it converges or diverges
and choose the most appropriate test listed, which you could use
to make the judgement.
a)
∞
∑
n
=
1
(
-
1
)
n
n
(
√
n
+
1
)
2
?
by the
?
b) Suppose that lim
n
→
∞
a
n
+
1
a
n
=
1
2
.
The series
∞
∑
n
=
1
n
3
a
n
?
by the ?
c)
∞
∑
n
=
1
(
n
!
)
3
(
3
n
)
!
?
by the ?
d)
∞
∑
n
=
2
1
n
p
ln
(
n
)
?
by the ?
e) Suppose that
∞
∑
n
=
1
a
n
converges and that
a
n
>
0 for all
n
.
The series
∞
∑
n
=
1
a
n
1
+
a
n
?
by the ?
f)
∞
∑
n
=
1
arctan
(
n
)
2
n
+
n
2
?
by the ?
Answer(s) submitted:
•
diverges
•
divergence test
•
converges
•
ratio test
•
converges
•
ratio test
•
diverges
•
integral test
•
converges
•
comparison test
•
converges
•
comparison test
submitted: (correct)
recorded: (correct)
Problem 4.
(1 point)
Prologue:
The series
∞
∑
n
=
1
r
n
n
p
converges when
-
1
<
r
<
1 and di-
verges when
|
r
|
>
1. This is true regardless of the value of the
constant
p
. When
r
=
1 the series is a
p
-series. It converges if
p
>
1 and diverges otherwise.
Problem:
Each of the series below can be compared to a series of
the form
∞
∑
n
=
1
r
n
n
p
. In each case, determine the best value of
r
and
decide whether the series converges.
A.
∞
∑
n
=
1
(
7
+
n
(
5
)
n
)
-
4
r
=
.
This series ?
.
B.
∞
∑
n
=
1
n
π
4
2
n
4
n
+
n
9
r
=
.
This series ?
.
C.
∞
∑
n
=
1
√
n
+
7
n
4
+
5
r
=
.
This series ?
.
D.
∞
∑
n
=
1
7
n
2
+
4
n
+
7
-
9
n
7
n
+
8
+
4
n
+
5
√
n
9
r
=
.
This series ?
.
Answer(s) submitted:
•
1
5
4
•
CONVERGES
•
4
•
DIVERGES
•
1
•
CONVERGES
•
1
7
9
•
CONVERGES
submitted: (correct)
recorded: (correct)
3
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Problem 5.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing “Submit”.
Test each series below for convergence.
a)
∞
∑
n
=
1
n
(
n
6
+
5
)
1
/
2
[?/Converges/Diverges]
b)
∞
∑
n
=
1
3
n
2
+
4
n
[?/Converges/Diverges]
c)
∞
∑
n
=
1
8
n
9
n
[?/Converges/Diverges]
d)
∞
∑
n
=
1
sin
6
n
[?/Converges/Diverges]
e)
∞
∑
n
=
1
n
5
+
4
n
1
+
9
[?/Converges/Diverges]
Answer(s) submitted:
•
Converges
•
Converges
•
Converges
•
Diverges
•
Diverges
submitted: (correct)
recorded: (correct)
Problem 6.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing “Submit”.
Test each series below for convergence.
a)
∞
∑
n
=
1
(
-
1
)
n
sin
2
n
[?/Converges/Diverges]
b)
∞
∑
n
=
1
cos
(
n
π
)
n
4
/
7
[?/Converges/Diverges]
c)
∞
∑
n
=
1
(
-
1
)
n
6
n
9
n
+
9
[?/Converges/Diverges]
d)
∞
∑
n
=
1
(
-
1
)
n
-
1
n
+
6
• ?
• Converges Absolutely
• Converges Conditionally
• Diverges
e)
∞
∑
n
=
1
(
-
8
)
n
n
4
• ?
• Converges Absolutely
• Converges Conditionally
• Diverges
Answer(s) submitted:
•
Converges
•
Converges
•
Diverges
•
Converges Conditionally
•
Diverges
submitted: (correct)
recorded: (correct)
4
Problem 7.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Each of the series given below converges.
Enter the one-letter
label of the FIRST correct reason for convergence.
A. Convergent geometric series
B. Convergent
p
series
C. Comparison (or Limit Comparison) with a geometric or
p
series
D. Alternating Series Test
E. None of the above
1.
∞
∑
n
=
1
(
n
+
1
)(
3
)
n
2
2
n
2.
∞
∑
n
=
1
4
(
7
)
n
9
2
n
3.
∞
∑
n
=
1
n
2
+
√
n
n
4
-
1
4.
∞
∑
n
=
1
cos
(
n
π
)
log
(
3
n
)
5.
∞
∑
n
=
1
sin
2
(
5
n
)
n
2
6.
∞
∑
n
=
1
(
-
1
)
n
4
n
+
2
Answer(s) submitted:
•
C
•
A
•
C
•
D
•
C
•
D
submitted: (correct)
recorded: (correct)
Problem 8.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
The three series
A
=
∑
a
n
,
B
=
∑
b
n
, and
C
=
∑
c
n
have terms
a
n
=
1
n
10
,
b
n
=
1
n
2
,
c
n
=
1
n
.
Use the Limit Comparison Test to compare each of the follow-
ing series to one of the series above. Enter one of the following
two-letter codes for each answer:
AC
,
BC
,
CC
,
AD
,
BD
,
CD
.
The first letter is the name of the series above used to make a valid
comparison; the second letter is C if the given series converges, or
D if it diverges.
1.
∞
∑
n
=
1
5
n
2
+
8
n
9
2
n
10
+
7
n
3
-
2
2.
∞
∑
n
=
1
2
n
2
+
n
10
1309
n
12
+
7
n
2
+
5
3.
∞
∑
n
=
1
8
n
6
+
n
2
-
8
n
7
n
16
-
5
n
12
+
7
Answer(s) submitted:
•
CD
•
BC
•
AC
submitted: (correct)
recorded: (correct)
5
Problem 9.
(1 point)
For each sequence
a
n
below, find a number
p
such that
n
p
a
n
has a
finite non-zero limit as
n
→
∞
.
(According to the limit comparison test, the convergence or diver-
gence of the
p
-series
∞
∑
n
=
1
1
n
p
then predicts the same behaviour for
the series
∞
∑
n
=
1
a
n
.)
A.
a
n
= (
4
+
2
n
)
-
2
p
=
B.
a
n
=
2
n
7
+
n
p
=
C.
a
n
=
6
n
2
+
2
n
+
6
7
n
2
+
7
n
+
2
p
=
D.
a
n
=
6
n
2
+
2
n
+
4
7
n
2
+
7
n
+
2
√
n
3
p
=
Answer(s) submitted:
•
2
•
7
•
0
•
0
submitted: (correct)
recorded: (correct)
Problem 10.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Consider the series
S
=
∞
∑
n
=
1
a
n
, with
a
n
= (
-
1
)
n
-
1
3
n
n
3
.
(a)
Evaluate the following limit. (Enter a number or one of the
3-letter codes ”inf” or ”div”.)
Answer: lim
n
→
∞
a
n
+
1
a
n
=
(b)
What does the Ratio Test say about the series
S
? Choose from
”Convergent”, ”Divergent”, or ”Inconclusive”, below.
Answer:
• choose one
• Convergent
• Divergent
• Inconclusive
(c)
Decide whether the series is absolutely convergent, condition-
ally convergent, or divergent. Use the menu below to present your
findings.
Answer:
• choose one
• Absolutely Convergent
• Conditionally Convergent
• Divergent
Answer(s) submitted:
•
3
•
Divergent
•
Divergent
submitted: (correct)
recorded: (correct)
6
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Problem 11.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Consider the series
S
=
∞
∑
n
=
1
a
n
, with
a
n
=
(
-
1
)
n
n
5
.
(a)
Evaluate the following limit. (Enter a number or one of the
3-letter codes ”inf” or ”div”.)
Answer: lim
n
→
∞
a
n
+
1
a
n
=
(b)
What does the Ratio Test say about the series
S
? Choose from
”Convergent”, ”Divergent”, or ”Inconclusive”, below.
Answer:
• choose one
• Convergent
• Divergent
• Inconclusive
(c)
Decide whether the series is absolutely convergent, condition-
ally convergent, or divergent. Use the menu below to present your
findings.
Answer:
• choose one
• Absolutely Convergent
• Conditionally Convergent
• Divergent
Answer(s) submitted:
•
1
•
Inconclusive
•
Absolutely Convergent
submitted: (correct)
recorded: (correct)
Problem 12.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Consider the series
S
=
∞
∑
n
=
1
a
n
, with
a
n
=
e
-
n
n
!.
(a)
Evaluate the following limit. (Enter a number or one of the
3-letter codes ”inf” or ”div”.)
Answer: lim
n
→
∞
a
n
+
1
a
n
=
(b)
What does the Ratio Test say about the series
S
? Choose from
”Convergent”, ”Divergent”, or ”Inconclusive”, below.
Answer:
• choose one
• Convergent
• Divergent
• Inconclusive
(c)
Decide whether the series is absolutely convergent, condition-
ally convergent, or divergent. Use the menu below to present your
findings.
Answer:
• choose one
• Absolutely Convergent
• Conditionally Convergent
• Divergent
Answer(s) submitted:
•
INF
•
Divergent
•
Divergent
submitted: (correct)
recorded: (correct)
7
Problem 13.
(1 point)
CAUTION:
You have five chances to submit an answer for this
problem. Make sure you have answered all the parts before press-
ing ”Submit”.
Consider the series
S
=
∞
∑
n
=
1
a
n
, with
a
n
= (
-
1
)
n
n
2
5
n
n
!
.
(a)
Evaluate the following limit. (Enter a number or one of the
3-letter codes ”inf” or ”div”.)
Answer: lim
n
→
∞
a
n
+
1
a
n
=
(b)
What does the Ratio Test say about the series
S
? Choose from
”Convergent”, ”Divergent”, or ”Inconclusive”, below.
Answer:
• choose one
• Convergent
• Divergent
• Inconclusive
(c)
Decide whether the series is absolutely convergent, condition-
ally convergent, or divergent. Use the menu below to present your
findings.
Answer:
• choose one
• Absolutely Convergent
• Conditionally Convergent
• Divergent
Answer(s) submitted:
•
0
•
Convergent
•
Absolutely Convergent
submitted: (correct)
recorded: (correct)
Problem 14.
(1 point)
For an alternating series whose summands are decreasing in mag-
nitude, the true sum
S
lies between any two successive partial
sums:
(
*
)
min
{
S
N
,
S
N
+
1
} ≤
S
≤
max
{
S
N
,
S
N
+
1
}
.
Consider
S
=
∞
∑
n
=
1
(
-
1
)
n
+
1
n
3
,
and
write
S
N
=
N
∑
n
=
1
(
-
1
)
n
+
1
n
3
.
(a)
Find the smallest value of
N
for which the interval bracketing
S
in line
(
*
)
above has length at most 10
-
3
.
Answer:
N
min
=
(b)
Using the
N
found in part
(a)
, approximate
S
by the midpoint
of the interval implicit in line
(
*
)
. A spreadsheet may be helpful
to calculate the sum
S
N
.
Answer:
S
≈
S
N
+
S
N
+
1
2
=
Answer(s) submitted:
•
9
•
0
.
901615
submitted: (correct)
recorded: (correct)
Problem 15.
(1 point)
Find all the values of x such that the given series would converge.
∞
∑
n
=
1
(
n
+
7
)
x
n
Answer:
Note:
Give your answer in
interval notation
.
Answer(s) submitted:
•
(
-
1
,
1
)
submitted: (correct)
recorded: (correct)
8
Problem 16.
(1 point)
Find all the values of x such that the given series would converge.
∞
∑
n
=
1
(
-
1
)
n
n
2
n
x
n
Answer:
Note:
Give your answer in
interval notation
.
Answer(s) submitted:
•
-
1
2
,
1
2
submitted: (correct)
recorded: (correct)
Problem 17.
(1 point)
For each
n
≥
2, let
a
n
=
n
3
+
7
n
6
n
2
+
3
n
+
1
sin
π
n
. Evaluate the fol-
lowing series:
S
=
∞
∑
n
=
2
a
n
+
1
-
a
n
a
n
a
n
+
1
=
Suggestion
: Flex your abstract thinking skills.
Work with the
given abstract form of
S
for as long as possible, so that your inter-
actions with the detailed definition of
a
n
can be laser-focused and
done last.
Answer(s) submitted:
•
31
22
-
6
π
submitted: (correct)
recorded: (correct)
Problem 18.
(1 point)
Given
p
(
t
) =
88
e
5
t
95
+
75
e
5
t
, consider the series
S
(
t
) =
∞
∑
n
=
1
e
-
p
(
t
)
log
(
n
)
.
Find the set of real numbers
t
for which
S
(
t
)
converges. Write
your answer in interval notation.
Interval for convergence:
Answer(s) submitted:
•
ln
95
13
5
,
∞
submitted: (correct)
recorded: (correct)
Problem 19.
(1 point)
Find the set of real numbers
z
for which the following series con-
verges:
S
(
z
) =
∞
∑
n
=
4
30
(
-
5
)
n
n
!
(
nz
)
n
.
Use interval notation to present your answer.
Set of real
z
values for convergence:
Hint
:
Remember that lim
n
→
∞
1
+
1
n
n
=
e
.
To treat borderline
z
-values, Stirling’s approximation of the factorial is accurate
enough:
n
!
≈
√
2
π
n
n
e
n
.
Answer(s) submitted:
•
5
e
,
∞
∪
-
∞
,
-
5
e
submitted: (correct)
recorded: (correct)
Problem 20.
(1 point)
Use interval notation to present the answers to these three ques-
tions.
(a)
Find the set of real numbers
t
for which the following series
converges:
A
(
t
) =
∞
∑
n
=
6
(
2
t
+
17
)
n
.
Answer:
(b)
Find the set of all real numbers
x
for which the following series
converges:
B
(
x
) =
∞
∑
n
=
5
arctan
-
61
x
29
n
.
Answer:
(c)
Find the largest interval including 0 on which the following
series converges:
C
(
θ
) =
∞
∑
n
=
16
2
n
sin
2
n
39
86
θ
.
Answer:
Answer(s) submitted:
•
(
-
9
,
-
8
)
•
tan
(
-
1
)
·
29
61
,
tan
(
1
)
·
29
61
•
(
-
1
.
7319
,
1
.
7319
)
submitted: (correct)
recorded: (correct)
9
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