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Bobbitt, James – Exam 3 – Due: Dec 4 2007, 11:00 pm – Inst: Diane Radin
1
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printout
should
have
18
questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
The due time is Central
time.
001
(part 1 oF 1) 10 points
Determine whether the sequence
{
a
n
}
con
verges or diverges when
a
n
= (

1)
n
µ
6
n
+ 5
7
n
+ 3
¶
,
and iF it does, fnd its limit.
1.
limit =
±
6
7
2.
limit = 0
3.
sequence diverges
correct
4.
limit =
5
3
5.
limit =
6
7
Explanation:
AFter division,
6
n
+ 5
7
n
+ 3
=
6 +
5
n
7 +
3
n
.
Now
5
n
,
3
n
→
0 as
n
→ ∞
,
so
lim
n
→∞
6
n
+ 5
7
n
+ 3
=
6
7
6
= 0
.
Thus as
n
→ ∞
, the values oF
a
n
oscillate be
tween values ever closer to
±
6
7
. Consequently,
the sequence diverges
.
keywords:
002
(part 1 oF 1) 10 points
Determine iF the sequence
{
a
n
}
converges
when
a
n
=
n
5
n
(
n

7)
5
n
,
and iF it does, fnd its limit
1.
sequence diverges
2.
limit =
e
35
correct
3.
limit =
e
7
5
4.
limit =
e

7
5
5.
limit = 1
6.
limit =
e

35
Explanation:
By the Laws oF Exponents,
a
n
=
µ
n

7
n
¶

5
n
=
µ
1

7
n
¶

5
n
=
h‡
1

7
n
·
n
i

5
.
But
‡
1 +
x
n
·
n
→
e
x
as
n
→ ∞
.
Consequently,
{
a
n
}
converges
and has
limit =
(
e

7
)

5
=
e
35
.
keywords: sequence, e, exponentials, limit
003
(part 1 oF 1) 10 points
Determine whether the series
2 + 3 +
9
2
+
27
4
+
···
is convergent or divergent, and iF convergent,
fnd its sum.
1.
convergent with sum =
1
4
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View Full Document Bobbitt, James – Exam 3 – Due: Dec 4 2007, 11:00 pm – Inst: Diane Radin
2
2.
divergent
correct
3.
convergent with sum = 9
4.
convergent with sum =
1
9
5.
convergent with sum = 4
Explanation:
The series
2 + 3 +
9
2
+
27
4
+
···
=
∞
X
n
=1
a r
n

1
is an infnite geometric series in which
a
= 2
and
r
=
3
2
. But such a series is
(i) convergent with sum
a
1

r
when

r

<
1,
(ii) divergent when

r
 ≥
1
.
Thus the given series is
divergent
.
keywords:
004
(part 1 oF 1) 10 points
Determine whether the series
∞
X
n
= 0
2 (cos
nπ
)
µ
1
3
¶
n
is convergent or divergent, and iF convergent,
fnd its sum.
1.
convergent with sum

2
3
2.
divergent
3.
convergent with sum

3
2
4.
convergent with sum
3
2
correct
5.
convergent with sum 3
6.
convergent with sum

3
Explanation:
Since
cos
nπ
= (

1)
n
,
the given series can be rewritten as an infnite
geometric series
∞
X
n
=0
2
µ

1
3
¶
n
=
∞
X
n
= 0
a r
n
in which
a
= 2
,
r
=

1
3
.
But the series
∑
∞
n
=0
ar
n
is
(i) convergent with sum
a
1

r
when

r

<
1,
and
(ii) divergent when

r
 ≥
1.
Consequently, the given series is
convergent with sum
3
2
.
keywords: geometric series, convergent
005
(part 1 oF 1) 10 points
Determine whether the infnite series
∞
X
n
=1
2(
n
+ 1)
2
n
(
n
+ 2)
converges or diverges, and iF converges, fnd
its sum.
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This note was uploaded on 09/14/2009 for the course M 408 L taught by Professor Cepparo during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Cepparo

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