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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 varies
 Calculus, Maclaurin Series, Power Series, Taylor Series

Click to edit the document details