TAYLOR SERIES
(10.3) (practice)
1. The following parts refer to
1
( )
3
f x
x
=
−
.
A. Write the Taylor series expansion for
( )
f x
about
x
= 0.
B. Find the Taylor series expansion for
( )
f x
about
x
= 2 without actually taking the derivatives. (Hint:
rewrite the denominator so that
2
x
−
appears.)
C. Use part A to find each of the following
(0)
f
′
(0)
f
′′′
)
0
(
)
5
(
f
D. Use part B to find each of the following
(2)
f
′
(2)
f
′′′
(5)
(2)
f
2. Find the exact value of the following sums:
A.
⋅
⋅
⋅
−
+
−
+
−
!
8
)
2
.
0
(
5
!
6
)
2
.
0
(
5
!
4
)
2
.
0
(
5
!
2
)
2
.
0
(
5
5
8
6
4
2
B.
⋅
⋅
⋅
+
−
+
−
!
7
)
2
.
0
(
!
5
)
2
.
0
(
!
3
4
)
2
.
0
(
)
2
.
0
(
8
6
2
C.
2
4
6
8
5(0.2)
5(0.2)
5(0.2)
5(0.2)
5
1!
2!
3!
4!
−
+
−
+
−⋅⋅⋅
D.
⋅
⋅
⋅
−
+
+
+
2
)
2
.
0
(
5
2
)
2
.
0
(
5
2
)
2
.
0
(
5
2
)
2
.
0
(
5
8
6
4
2
3. Use series to evaluate the limit
2
0
arctan( )
lim
1
x
x
x
x
e
→
⋅
−
.
4. Expand
2
2
)
(
h
R
mgR
F
+
=
in terms of
R
h
. Assume
R
is very large when compared to
h
.
5. Expand
(
)
R
a
R
Q
−
+
=
2
2
2
πσ
in terms of
R
a
. Assume
R
is very large when compared to
a
.

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