
Infinite Sequences and Series 17-31
This infinite power series is called Taylor’s series.
Thus, Taylor’s series converges for all values of x for which R
n
approaches 0 as n increases without
limit.
Remark 17.2 The above infinite series represents the function for only those values of x for which R
n
approaches zero as the number of terms gets larger and larger.
17.8.1 Maclaurin’s Series
If we take a = 0 in Taylor’s series, then we obtain the Taylor’s series about 0:
f x
f
n
x f x f '
x
f ''
x
f '''
n
n
n
( )
( )
!
( ) ( )
!
( )
!
( )= =
∑
+ + + +
=
∞
0
0 0
2
0
3
0
0
2 3
The above series is called Maclaurin’s series.
Example 17.44 If f(x) = e
x
, then find the Taylor’s series ...