010
10.0 points
Determine whether the following series
∞
s
n
=1
p
3

n
2
3 + 5
n
2
P
n
is absolutely convergent, conditionally con
vergent, or divergent.
1.
conditionally convergent
2.
divergent
3.
absolutely convergent
correct
Explanation:
The given series has the form
∞
s
n
=1
a
n
where
a
n
=
p
3

n
2
3 + 5
n
2
P
n
.
But then

a
n

1
/n
=
v
v
v
v
3

n
2
3 + 5
n
2
v
v
v
v
.
Thus
lim
n
→∞

a
n

1
/n
=
1
5
<
1
.
Consequently, by the Root Test, the given
series is
absolutely convergent
.
011
10.0 points
Determine whether the following series
4
5
+
4
·
7
5
·
7
+
4
·
7
·
10
5
·
7
·
9
+
4
·
7
·
10
·
13
5
·
7
·
9
·
11
+
···
is absolutely convergent, conditionally con
vergent, or divergent.
1.
divergent
correct
2.
absolutely convergent
choice (hac762) – HW Quest Week 8 – cepparo – (53850)
6
3.
conditionally convergent
Explanation:
One way to do this problem is to observe
that the numerators of the sequence which is
being summed are given recursively by
p
1
= 4
,
p
n
+1
=
p
n
(4 + 3
n
)
,
while the denominators are given by
q
1
= 5
,
q
n
+1
=
q
n
(5 + 2
n
)
.
The sequence we are interested in summing is
a
n
=
p
n
q
n
,
which by these recursion relations can be writ
ten
a
n
+1
=
p
n
+1
q
n
+1
=
p
n
(4 + 3
n
)
q
n
(5 + 2
n
)
=
a
n
p
4 + 3
n
5 + 2
n
P
.
Thus
lim
n
→∞
v
v
v
v
a
n
+1
a
n
v
v
v
v
= lim
n
→∞
4 + 3
n
5 + 2
n
=
3
2
>
1
,
so by the Ratio Test the original series is
divergent
.
012
10.0 points
Determine whether the series
∞
s
n
=1
2
·
4
·
6
·
. . .
·
(2
n
)
n
!
is absolutely convergent, conditionally con
vergent, or divergent.
1.
absolutely convergent
2.
conditionally convergent
3.
divergent
correct
Explanation:
∞
s
n
=1
2
·
4
·
6
·
. . .
·
(2
n
)
n
!
=
∞
s
n
=1
(2
·
1)
·
(2
·
2)
·
(2
·
3)
·
. . .
·
(2
n
)
n
!
=
∞
s
n
=1
2
n
n
!
n
!
=
∞
s
n
=1
2
n
It diverges by the divergence test.
013
10.0 points
Which, if either, of the following statements
are true?
A.
The Ratio Test can be used to deter
mine whether the series
∞
s
n
= 1
1
n
2
converges or diverges.
B. The Root Test can be used to determine
whether the series
∞
s
k
= 1
p
ln(
k
)
2 +
k
P
k
converges or diverges.
1.
A only
2.
B only
correct
3.
both of them
4.
neither of them
Explanation:
A. False: when
a
n
= 1
/n
2
, then
v
v
v
v
a
n
+1
a
n
v
v
v
v
=
p
n
n
+ 1
P
2
→
1
as
n
→ ∞
, so the Ratio Test is inconclusive.
choice (hac762) – HW Quest Week 8 – cepparo – (53850)
7
B. True: when
a
k
=
p
ln(
k
)
2 +
k
P
k
,
then

a
k

1
/k
=
ln(
k
)
2 +
k
→
0
as
k
→ ∞
, so
∑
a
k
is convergent by the Root
Test.
014
10.0 points
If lim
n
→∞
a
n
= 0, which, if any, of the
following statements are true:
(A)
lim
n
→∞
(
a
n
)
2
= 0
,
(B)
s
n
a
n
is convergent
.
1.
neither A nor B
2.
A only
correct
3.
B only
4.
both A and B
Explanation:
(A) TRUE: by Properties of Limits,
lim
n
→∞
(
a
n
)
2
=
±
lim
n
→∞
a
n
²
2
= 0
,
(B) FALSE: when
a
n
=
1
/n
,
then
lim
n
→∞
a
n
= 0, but
∞
s
n
= 1
a
n
=
∞
s
n
=1
1
n
diverges by the Integral Test.
015
10.0 points
Determine which, if any, of the series
A.