Pham, Quoc – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Eric Katerman
1
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20
questions.
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beFore answering.
The due time is Central
time.
001
(part 1 oF 1) 10 points
Which, iF any, oF the Following statements
are true?
A. IF 0
≤
a
n
≤
b
n
and
X
b
n
diverges, then
X
a
n
diverges
B. The Ratio Test can be used to determine
whether
X
1
/n
3
converges.
C. IF
X
a
n
converges, then
lim
n
→∞
a
n
= 0.
1.
A and B only
2.
A and C only
3.
B and C only
4.
A only
5.
B only
6.
C only
correct
7.
none oF them
8.
all oF them
Explanation:
A. ±alse: set
a
n
=
1
n
2
,
b
n
=
1
n
.
Then 0
≤
a
n
≤
b
n
, but the Integral Test
shows that
X
a
n
converges while
X
b
n
diverges.
B. ±alse: when
a
n
= 1
/n
3
, then
f
f
f
f
a
n
+1
a
n
f
f
f
f
=
n
3
(
n
+ 1)
3
→
1
as
n
→
,
∞
, so the Ratio Test is inconclu
sive.
C. True. To say that
X
a
n
converges is to
say that the limit
lim
n
→∞
s
n
oF its partial
sums
s
n
=
a
1
+
a
2
+
...
+
a
n
converges. But then
lim
n
→∞
a
n
=
s
n

s
n

1
= 0
.
keywords:
002
(part 1 oF 1) 10 points
Which one oF the Following properties does
the series
∞
X
n
= 3
(

1)
n

1
n

2
n
2
+
n

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

1)
n

1
n

1
n
2
+
n

3
=
∞
X
n
= 3
(

1)
n

1
f
(
n
)
where
f
is defned by
f
(
x
) =
x

2
x
2
+
x

3
.
Notice that
x
2
+
x

3
>
0 on [3
,
∞
), so the
terms in the given series are defned For all
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View Full DocumentPham, Quoc – Homework 13 – Due: Apr 24 2007, 3:00 am – Inst: Eric Katerman
2
n
≥
3.
On the other hand,
x

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

3)

(
x

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

3)
2
=

x
2

4
x
+ 1
(
x
2
+
x

3)
2
;
in particular,
f
is decreasing on [5
,
∞
). Thus
by the Limit Comparison Test and the
p
series
Test with
p
= 1, we see that the series
∞
X
n
= 5
f
(
n
)
diverges, so the given series fails to be abso
lutely convergent. But
n
≥
5
=
⇒
f
(
n
)
> f
(
n
+ 1)
,
while
lim
x
→∞
f
(
x
) = 0
.
Consequently, by The Alternating Series Test,
the given series is
conditionally convergent
.
keywords:
003
(part 1 of 1) 10 points
Determine which, if any, of the series
A.
1 +
1
2
+
1
4
+
1
8
+
1
16
+
...
B.
∞
X
m
= 3
m
+ 3
m
2
ln
m
+ 2
are divergent.
1.
B only
correct
2.
both of them
3.
A only
4.
neither of them
Explanation:
A. Convergent:
given series is a geometric
series
∞
X
n
= 0
ar
n
with
a
= 1 and
r
=
1
2
<
1.
B. Divergent:
use Limit Comparison Test
and Integral Test with
f
(
x
) =
1
x
ln
x
.
keywords:
004
(part 1 of 1) 10 points
Which of the following properties does the
series
∞
X
m
= 1
(

5)
m
+1
4
2
m
have?
1.
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 Mathematical Series, lim

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