Math 153  Spring 2010
NAME
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Exam 1A
7 April 2010
Answer the following questions. The answers must be clear, intelligible, and you must show
your work. Provide explanation for all your steps. Your grade will be determined by
adherence to these criteria. Use of books, notes and calculators is strictly forbidden. The
point value of each problem is given in the left hand margin.
(8 pts.)
1.
Determine whether the sequence converges or diverges. If it converges, ±nd the
limit.
a
n
=
9
n
+1
10
n
;
n
±
1
We can rewrite
a
n
= 9
²
±
9
10
²
n
and therefore
lim
n
!1
a
n
= lim
n
!1
9
²
±
9
10
²
n
= 9
²
lim
n
!1
±
9
10
²
n
= 9
²
0
The sequence converges.
1
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View Full Document2.
You are given the sequence
f
a
n
g
with:
a
n
=
n
±
3
5
n
+ 1
;
n
²
1
Answer the following questions about this sequence.
(4 pts.) a). Is the sequence increasing or decreasing?
We consider the function:
f
(
x
) =
x
±
3
5
x
+ 1
for all
x
²
1. The ±rst derivative of
f
(
x
) is
f
0
(
x
) =
(5
x
+ 1)
±
5
³
(
x
±
3)
(5
x
+ 1)
2
=
16
(5
x
+ 1)
2
which is always positive. Therefore
f
(
x
) is increasing and the sequence
f
a
n
g
is also increasing.
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