Portillo, Tom – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi
1
This printout should have 17 questions.
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beFore answering.
The due time is Central
time.
001
(part 1 oF 1) 10 points
Determine iF the sequence
{
a
n
}
converges
when
a
n
=
1
n
ln
µ
4
5
n
+ 5
¶
,
and iF it does, fnd its limit.
1.
limit = ln
4
5
2.
the sequence diverges
3.
limit = ln
2
5
4.
limit =

ln5
5.
limit = 0
correct
Explanation:
AFter division by
n
we see that
4
5
n
+ 5
=
4
n
5 +
5
n
,
so by properties oF logs,
a
n
=
1
n
ln
4
n

1
n
ln
µ
5 +
5
n
¶
.
But by known limits (or use L’Hospital),
1
n
ln
4
n
,
1
n
ln
µ
5 +
5
n
¶
→
0
as
n
→ ∞
. Consequently, the sequence
{
a
n
}
converges and has
limit = 0
.
keywords: limit, sequence, log Function,
002
(part 1 oF 1) 10 points
Determine whether the sequence
{
a
n
}
con
verges or diverges when
a
n
=
14
n
2
7
n
+ 1

2
n
2
+ 3
n
+ 1
,
and iF it does, fnd its limit
1.
limit =
4
7
2.
limit = 0
3.
limit =
6
7
4.
the sequence diverges
5.
limit =
12
7
correct
Explanation:
AFter bringing the two terms to a common
denominator we see that
a
n
=
14
n
3
+ 14
n
2

(7
n
+ 1)
(
2
n
2
+ 3
)
(7
n
+ 1)(
n
+ 1)
=
12
n
2

21
n

3
7
n
2
+ 8
n
+ 1
.
Thus
a
n
=
12

21
n

3
n
2
7 +
8
n
+
1
n
2
.
But
21
n
,
3
n
2
,
8
n
,
1
n
2
→
0
as
n
→ ∞
.
Thus
{
a
n
}
converges and has
limit =
12
7
.
keywords: sequence, convergence
003
(part 1 oF 1) 10 points
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View Full DocumentPortillo, Tom – Homework 10 – Due: Nov 7 2006, 3:00 am – Inst: David Benzvi
2
Determine if the sequence
{
a
n
}
converges
when
a
n
=
n
3
n
(
n

2)
3
n
,
and if it does, Fnd its limit
1.
sequence diverges
2.
limit =
e
2
3
3.
limit = 1
4.
limit =
e

6
5.
limit =
e
6
correct
6.
limit =
e

2
3
Explanation:
By the Laws of Exponents,
a
n
=
µ
n

2
n
¶

3
n
=
µ
1

2
n
¶

3
n
=
h‡
1

2
n
·
n
i

3
.
But
‡
1 +
x
n
·
n
→
e
x
as
n
→ ∞
.
Consequently,
{
a
n
}
converges
and has
limit =
(
e

2
)

3
=
e
6
.
keywords: sequence, e, exponentials, limit
004
(part 1 of 1) 10 points
Determine whether the sequence
{
a
n
}
con
verges or diverges when
a
n
=
(

1)
n

1
n
n
2
+ 3
,
and if it converges, Fnd the limit.
1.
converges with limit = 0
correct
2.
converges with limit =

1
3
3.
sequence diverges
4.
converges with limit = 3
5.
converges with limit =
1
3
6.
converges with limit =

3
Explanation:
After division,
a
n
=
(

1)
n

1
n
n
2
+ 3
=
(

1)
n

1
n
+
1
n
.
Consequently,
0
≤ 
a
n

=
1
n
+
1
n
≤
1
n
.
But 1
/n
→
0 as
n
→ ∞
, so by the Squeeze
theorem,
lim
n
→∞

a
n

= 0
.
But

a
n
 ≤
a
n
≤ 
a
n

,
so by the Squeeze theorem again the given
sequence
{
a
n
}
converges and has
limit = 0
.
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 Spring '06
 Benzvi
 Limit, Portillo, David Benzvi

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