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Math 141, Sections 03**, T Pilachowski, Spring 2007
April 27, 2007
MATH 141 – TEST 4
(9.1 – 9.9)
[Pilachowski]
1. a. (12 points) Use derivatives to find the 4th Taylor polynomial
( )
x
p
4
about
x
= 0 for the function
()
x
e
x
f
2
=
. (You must show your steps to receive full credit.)
() ()
()
()
( )
( )
( )
( )
()
( )
()
()
()
()
()
()
()
()
16
0
16
8
0
8
4
0
4
2
0
2
1
0
4
2
4
3
2
3
2
2
2
1
2
1
2
=
⇒
=
=
⇒
=
=
⇒
=
=
⇒
=
=
⇒
=
f
e
x
f
f
e
x
f
f
e
x
f
f
e
x
f
f
e
x
f
x
x
x
x
x
()
4
3
2
4
3
2
4
3
2
3
4
2
2
1
4
16
3
8
2
4
2
1
!
!
!
x
x
x
x
x
x
x
x
x
p
+
+
+
+
=
+
+
+
+
=
b. (10 points) Find the sum of the series
∑
∞
=
+
+
0
3
2
1
2
3
n
n
n
.
( )
2
3
4
8
3
1
8
3
1
8
3
4
3
8
3
2
3
8
3
2
3
2
3
2
3
4
1
4
3
0
4
3
0
0
2
0
2
3
0
3
2
1
=
∗
=
∗
=
−
∗
=
⎟
⎠
⎞
⎜
⎝
⎛
∗
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∗
=
∗
=
∑
∑
∑
∑
∞
=
∞
=
∞
=
∞
=
+
+
n
n
n
n
n
n
n
n
n
n
2. a. (12 points) Determine whether the series
∑
∞
=
+
−
1
5
2
n
n
n
converges. (You must show work that supports
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This note was uploaded on 09/12/2009 for the course MATH 141 taught by Professor Hamilton during the Spring '07 term at Maryland.
 Spring '07
 Hamilton
 Derivative

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