1. (12 points)
For each sequence below either calculate the limit or write “does not converge”:
a.
lim
n

>
∞
n
3
(2
n

1)
3
ANS: This converges to
1
8
as this is the ratio of the coefficients of the
leading terms and the polynomials have the same degree.
b.
lim
n

>
∞
ne

3
n
ANS: This converges to 0. We can think of
e

3
n
as either exponential
decay, or as an exponential function on the bottom of a fraction.
c.
lim
n

>
∞
(

1)
n
+
1
n
ANS: The first term
(

1)
n
oscillates, and the second term goes to zero,
so their sum doesn’t converge.
d.
lim
n

>
∞
sin(
(

1)
n
n
)
ANS: This converges to 0. To do this one you had to remember that:
•
lim
n

>
∞
sin(
(

1)
n
n
) = sin(lim
n

>
∞
(

1)
n
n
)
because sin is continuous.
•
lim
n

>
∞
(

1)
n
n
= 0
.
•
sin(0) = 0
.
1
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2. (14 points)
Place each sequence in the appropriate place on the following list by either
•
putting the sequence in the same box as a sequence that has the same rate
of growth
•
putting a sequence between two boxes, to say that its rate of growth is slower
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 Spring '09
 COSTELLO
 Math, Exponential Function, Polynomials, Mathematical Induction, Natural number

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