adams (bda255) – HW14 – Radin – (56635)
1
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beFore answering.
001
10.0 points
Which, iF any, oF the Following statements
are true?
A. IF
s
a
n
is divergent, then
s

a
n

is
divergent.
B. The Ratio Test can be used to determine
whether
s
1
/n
! converges.
C. IF lim
n
→∞
a
n
= 0, then
s
a
n
converges.
1.
A and C only
2.
none oF them
3.
A only
4.
C only
5.
B only
6.
A and B only
correct
7.
all oF them
8.
B and C only
Explanation:
A. True: iF
s

a
n

were convergent, then
s
a
n
would be absolutely convergent,
hence convergent.
B. True: when
a
n
= 1
/n
!, then
v
v
v
v
a
n
+1
a
n
v
v
v
v
=
1
n
+ 1
→
0
as
n
→
,
∞
, so
s
a
n
is convergent by
Ratio Test.
C. ±alse: 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.
002
10.0 points
Which one oF the Following properties does
the series
∞
s
k
= 2
(

1)
k
−
1
k

3
k
2
+
k

4
have?
1.
absolutely convergent
2.
conditionally convergent
correct
3.
divergent
Explanation:
The given series has the Form
∞
s
k
= 2
(

1)
k
−
1
k

1
k
2
+
k

4
=
∞
s
k
= 2
(

1)
k
−
1
f
(
k
)
where
f
is defned by
f
(
x
) =
x

3
x
2
+
x

4
.
Notice that
x
2
+
x

4
>
0 on [2
,
∞
), so the
terms in the given series are defned For all
k
≥
2.
On the other hand,
x

3
>
0 on
(3
,
∞
), so
x >
3
=
⇒
f
(
x
)
>
0
.
Now, by the Quotient Rule,
f
′
(
x
) =
(
x
2
+
x

4)

(
x

3)(2
x
+ 1)
(
x
2
+
x

4)
2
=

x
2

6
x
+ 1
(
x
2
+
x

4)
2
;
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View Full Documentadams (bda255) – HW14 – Radin – (56635)
2
in particular,
f
is decreasing on [7
,
∞
). Thus
by the Limit Comparison Test and the
p
series
Test with
p
= 1, we see that the series
∞
s
k
= 7
f
(
k
)
diverges, so the given series fails to be abso
lutely convergent. But
k
≥
7
=
⇒
f
(
k
)
> f
(
k
+ 1)
,
while
lim
x
→∞
f
(
x
) = 0
.
Consequently, by The Alternating Series Test,
the given series is
conditionally convergent
.
003
10.0 points
Determine which, if any, of the series
A. 1 +
1
2
+
1
4
+
1
8
+
1
16
+
. . .
B.
∞
s
m
= 3
m
+ 2
(
m
ln
m
)
2
are convergent.
1.
both of them
correct
2.
B only
3.
A only
4.
neither of them
Explanation:
A. Convergent: given series is a geometric se
ries
∞
s
n
= 0
ar
n
with
a
= 1 and
r
=
1
2
<
1.
B. Convergent: use Limit Comparison Test
and Integral Test with
f
(
x
) =
1
x
(ln
x
)
2
.
004
10.0 points
Determine whether the series
∞
s
m
= 1
(

3)
m
+1
2
3
m
is absolutely convergent, conditionally con
vergent or divergent.
1.
conditionally convergent
2.
divergent
3.
absolutely convergent
correct
Explanation:
The given series can be written in the form
∞
s
m
= 1
(

3)
m
+1
2
3
m
=
∞
s
m
= 1
a
n
with
a
m
= (

1)
m
+1
3
m
+1
2
3
m
= (

1)
m
+1
3
p
3
2
3
P
m
.
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 Spring '09
 RAdin
 Calculus, Mathematical Series, lim

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