Matthew Choi
Math6B01W15CHEN
Assignment HW4 due 02/05/2015 at 11:59pm PST
1.
(1 pt) Use the limit comparison test to determine whether
∞
∑
n
=
16
a
n
=
∞
∑
n
=
16
6
n
3

6
n
2
+
16
6
+
3
n
4
converges or diverges.
(a) Choose a series
∞
∑
n
=
16
b
n
with terms of the form
b
n
=
1
n
p
and apply the limit comparison test.
Write your answer as a
fully simplified fraction. For
n
≥
16,
lim
n
→
∞
a
n
b
n
=
lim
n
→
∞
(b) Evaluate the limit in the previous part. Enter
∞
as
infinity
and

∞
as
infinity.
If the limit does not exist, enter
DNE.
lim
n
→
∞
a
n
b
n
=
(c) By the limit comparison test, does the series converge, di
verge, or is the test inconclusive?
?
Correct Answers:
•
6*nˆ46*nˆ3+16*n
•
3*nˆ4+6
•
2
•
Diverges
2.
(1 pt) Each of the following statements is an attempt to
show that a given series is convergent or divergent not using the
Comparison Test (NOT the Limit Comparison Test.) For each
statement, enter C (for ”correct”) if the argument is valid, or
enter I (for ”incorrect”) if any part of the argument is flawed.
(Note: if the conclusion is true but the argument that led to it
was wrong, you must enter I.)
1. For all
n
>
1,
1
n
ln
(
n
)
<
2
n
, and the series 2
∑
1
n
diverges,
so by the Comparison Test, the series
∑
1
n
ln
(
n
)
diverges.
2. For all
n
>
1,
n
5

n
3
<
1
n
2
, and the series
∑
1
n
2
converges,
so by the Comparison Test, the series
∑
n
5

n
3
converges.
3. For all
n
>
2,
1
n
2

4
<
1
n
2
, and the series
∑
1
n
2
converges,
so by the Comparison Test, the series
∑
1
n
2

4
converges.
4. For all
n
>
1,
arctan
(
n
)
n
3
<
π
2
n
3
, and the series
π
2
∑
1
n
3
con
verges, so by the Comparison Test, the series
∑
arctan
(
n
)
n
3
converges.
5. For all
n
>
2,
ln
(
n
)
n
>
1
n
, and the series
∑
1
n
diverges, so
by the Comparison Test, the series
∑
ln
(
n
)
n
diverges.
6. For all
n
>
2,
ln
(
n
)
n
2
>
1
n
2
, and the series
∑
1
n
2
converges,
so by the Comparison Test, the series
∑
ln
(
n
)
n
2
converges.
Correct Answers:
•
I
•
I
•
I
•
C
•
C
•
I
3.
(1 pt)
Consider the series
∞
∑
n
=
1
1
n
(
n
+
3
)
Determine whether the series converges, and if it converges,
determine its value.
You've reached the end of your free preview.
Want to read all 4 pages?
 Fall '13
 Polynomials, Mathematical Series, lim