Math 12
Tufts University
Department of Mathematics
Exam 3
November 15, 2004
11:50–1:20
No books, notes,
or calculators
. Cross out what you do not want us to grade. You must
give reasons for full credit. You are required to sign
your exam book. With your signature
you are pledging that you have neither given nor received assistance on this exam.
You may assume known the facts below;
R
is the radius of convergence.
e
x
=
∞
n
=0
x
n
n
!
, R
=
∞
1
1

x
=
∞
n
=0
x
n
,
R
= 1
1
.
(12 points)
For each of the following power series, find the radius of convergence and
the interval of convergence.
(a)
∞
n
=0
8
n
n
x
3
n
(b)
∞
n
=0
(2
n
+ 1)!
3
n
(
x
+ 2)
n
.
2
.
(20 points)
Using the definition of a Taylor series, find the Taylor series for
f
(
x
)
centered at
a
= 1 for the following functions
f
(
x
). [Do not use other known series for your
derivation].
(a)
f
(
x
) =
x
3

2
x
+ 1
(b)
f
(
x
) =
1
(2
x
+ 1)
2
.
3
.
(12 points)
Using an infinite series given at the start of the exam,
(a) evaluate
∞
n
=0
(ln 2)
n
n
!
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 Spring '08
 GARANT
 Math, Power Series, Taylor Series, Mathematical Series, n=0

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