212 Chapter 7Sequences and series
(iii)
For example, ifx 1 = 11 , using the first five terms of the series,
≈ 1 × 1 1.06065941 1 ≈ 1 2.999998
- Trigonometric functions
The trigonometric functions sin 1 x, cos 1 xand tan 1 xhave continuous derivatives at
x 1 = 10 , and can be expanded as MacLaurin series. For example, for the sine function,
f(x) 1 = 1 sin 1 x, f′(x) 1 = 1 cos 1 x, f′′(x) 1 = 1 −sin 1 x, f′′′(x) 1 = 1 −cos 1 x, -
f(0) 1 = 1 0, f′(0) 1 = 1 1, f′′(0) 1 = 1 0, f′′′(0) 1 = 1 −1, -
Then
Comparison with the exponential series shows that the sine series converges for all
values of x.
- The logarithmic function
The function ln 1 xcannotbe expanded as a power series in xbecause the function
and all its derivatives are discontinuous atx 1 = 10. However, the functionln(1 1 + 1 x)is
well behaved atx 1 = 10 :
f(x) 1 = 1 ln(1 1 + 1 x) f(0) 1 = 10
f′(x) f′(0) 1 = 11
f′′(x) f′′(0) 1 = 1 − 1
f′′′(x) f′′′(0)= 12
f′′′′(x) f′′′′(0) 1 = 1 −3!
--
=
−
()+x
6
1
4=
()+x
2
1
3=
−
()+x
1
1
2=
+x
1
1
sinxx
xxx
=−
!
!
−
!
357357
8
981
1
16
1
512
1
8192
5
524288
=+−+− +
=+−+− +
81
16 512 8192
5
524288
23 4xx x x
()
−
()
−
()
121232338!
x
()881
8
81
1
28
1212 1+=+
=+
x +
xx
2212228
()
−
()
!
x