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MAC 2312
Exam 3 Solutions
July 26, 2011
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{z
}
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Read all of the following information before starting the exam:
•
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take oﬀ points if I cannot see how you arrived at your answer (even if your ﬁnal answer is
correct).
•
Circle or otherwise indicate your ﬁnal answers.
•
This test has 5 problems and is worth 50 points. It is your responsibility to make sure
that you have all of the pages!
•
Since time is limited, it is crucial that you
think
before you compute.
•
You must state any theorems or tests that you use. All hypotheses must be veriﬁed.
•
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Recall that if

x

<
1, then
∞
X
n
=0
x
n
=
1
1

x
.
a) Find a power series representation for the function
f
(
x
) =
1
1 +
x
2
. What is its radius of
convergence?
Solution:
By the above, we have
1
1 +
x
2
=
1
1

(

x
2
)
=
∞
X
n
=0
(

x
2
)
n
=
∞
X
n
=0
(

1)
n
x
2
n
.
This series is geometric and is convergent if

x
2

<
1, hence

x

<
1, so the radius of convergence
is
R
= 1.
b) Use part (a) to ﬁnd the Maclaurin series for arctan
x
. What is its radius of convergence?
(Hint: Recall that
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This note was uploaded on 10/13/2011 for the course MAC 2312 taught by Professor Bonner during the Summer '08 term at University of Florida.
 Summer '08
 Bonner
 Calculus

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