214 Chapter 7Sequences and series
(7.24)
This power series in(x 1 − 1 a)is called a Taylor series expansionof the functionf(x).
The verification and conditions of existence of the Taylor series are as for the
MacLaurin series.
EXAMPLE 7.13Taylor series
(i) Expandx
4about the pointx 1 = 11.
We have
f(x) 1 = 1 x
4, f′(x) 1 = 14 x
3, f′′(x) 1 = 141 × 13 x
2, f′′′(x) 1 = 1 4!x, f′′′′(x) 1 = 1 4!,
f
(n)(x) 1 = 1 0 if n 1 > 14
f′′′′(1) 1 =4!
Therefore,
(ii) Expand cos 1 xaboutx 1 = 1 π 22.
f(x) 1 = 1 cos 1 x, f′(x) 1 = 1 −sin 1 x, f′′(x) 1 = 1 −cos 1 x, f′′′(x) 1 = 1 sin 1 x, =
f(π 2 2) 1 = 1 0, f′(π 2 2) 1 = 1 −1, f′′(π 2 2) 1 = 1 0, f′′′(π 2 2) 1 = 1 1, =
Therefore
0 Exercises 68–71
7.7 Approximate values and limits
The MacLaurin and Taylor series provide a systematic tool for approximating
functions in the form of polynomials. Consider, for example, the logarithmic series
ln( ) 1
234
11
234+=−+−+, −<≤+xx
xxx
x
cos ( )
!!!
xx=−−+−−−+−ππ π π2( 2)( 2) ( 2x x x
351
3
1
5
1
7
))
7+
xxx
421
4
1
1
4
2
1
4
=+ − + − +
() ()
33
11
34
()()xx−+−
ff f f() , ()
!
!
,()
!
!
,()
!
!
11 1 ,
4
3
1
4
2
1
4
1
= ′ = ′′ = ′′′ =
=
−
!
=∑
nnnxa
n
fa
0∞()
()
()fx fa
xa
fa
xa
fa
xa
() ()
()
()
()
()
()
=+
−
!
′ +
−
!
′′ +
−
12
2 333!
fa′′′()+