11.10 Taylor and Maclaurin Series
•
Find the Taylor series (or Maclaurin Series) of a function
–
Calculate the coeﬃcients directly by using
Theorem 5
.
Theorem 5
If
f
has a power series representation (expansion) at
a
, that is, if
f
(
x
) =
∞
X
n
=0
c
n
(
x

a
)
n

x

a

< R
then its coeﬃcients are given by the formula
c
n
=
f
(
n
)
(
a
)
n
!
–
Use some Taylor series (or Maclaurin Series) we have already
known.
*
Some Important Maclaurin Series.
1
1

x
=
∞
X
n
=0
x
n
= 1 +
x
+
x
2
+
x
3
+
···
(

1
,
1)
e
x
=
∞
X
n
=0
x
n
n
!
= 1 +
x
1!
+
x
2
2!
+
x
3
3!
+
···
(
∞
,
∞
)
sin
x
=
∞
X
n
=0
(

1)
n
x
2
n
+1
(2
n
+ 1)!
=
x

x
3
3!
+
x
5
5!

x
7
7!
+
···
(
∞
,
∞
)
cos
x
=
∞
X
n
=0
(

1)
n
x
2
n
(2
n
)!
= 1

x
2
2!
+
x
4
4!

x
6
6!
+
···
(
∞
,
∞
)
arctan
x
=
∞
X
n
=0
(

1)
n
x
2
n
+1
2
n
+ 1
=
x

x
3
3
+
x
5
5

x
7
7
+
···
[

1
,
1]
*
Multiplication and Division: the ﬁrst few terms.
1
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 Spring '08
 varies
 Calculus, Maclaurin Series, Power Series, Taylor Series, §11.10 Taylor

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