Thus the integral, and the sum, converge.
B.
Again, we note that
a
n
=
n
+
3
n
2
+
6
n
+
3
is a sequence of positive
decreasing terms, so that the Integral Test applies. We compare
∞
∑
n
=
1
n
+
3
n
2
+
6
n
+
3
with
R
∞
c
n
+
3
n
2
+
6
n
+
3
dn
. Taking
c
=
1, we have
Z
∞
1
n
+
3
n
2
+
6
n
+
3
dn
=
lim
n
→
∞
1
2
ln
(
b
2
+
6
b
+
3
)

1
2
ln
(
10
)
.
Because lim
n
→
∞
ln
(
b
2
+
6
b
+
3
)
diverges, we know that the integral,
and thus the sum, diverge.
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
Correct Answers:
•
1/6ˆn
•
1/(6*ln(6))
•
A
•
(n+3)/(nˆ2 + 6 n + 3)
•
diverges
•
B
4.
(1 point)
For each of the following series, indicate whether the integral
test can be used to determine its convergence or not, and if not,
why.
A.
∞
∑
n
=
1
ln
(
0
.
5
n
)
Can the integral test be used to test convergence?
•
A. no, because the terms in the series do not decrease
in magnitude
•
B. no, because the terms in the series are not all positive
for
n
≥
c
, for some
c
>
0
•
C. no, because the series is not a geometric series
•
D. no, because the terms in the series are not recursively
defined
•
E. no, because the terms in the series are not defined for
all
n
•
F. yes
B.
∞
∑
n
=
1
cos
(
n
)
n
2
Can the integral test be used to test convergence?
•
A. no, because the terms in the series do not decrease
in magnitude
•
B. no, because the terms in the series are not all positive
for
n
≥
c
, for some
c
>
0
•
C. no, because the series is not a geometric series
•
D. no, because the terms in the series are not recursively
defined
•
E. no, because the terms in the series are not defined for
all
n
•
F. yes
Solution:
SOLUTION
Recall that the integral test can be used when we consider
∑
a
n
and the function
a
n
=
f
(
n
)
is decreasing and positive for
n
≥
c
. (Where
c
is some number that is fixed for the given
f
(
n
)
).
Thus, for
∞
∑
n
=
1
ln
(
0
.
5
n
)
, the integral test cannot be used, be
cause the terms in the series do not decrease in magnitude, and
for
∞
∑
n
=
1
cos
(
n
)
n
2
, the integral test cannot be used, because the terms
in the series are not all positive for
n
≥
c
, for some
c
>
0.
Answer(s) submitted:
•
•
(incorrect)
Correct Answers:
•
A
•
B
5.
(1 point) Compute the value of the following improper
integral. If it converges, enter its value. Enter
infinity
if it di
verges to
∞
, and
infinity
if it diverges to

∞
. Otherwise, enter
diverges.