210 Chapter 7Sequences and series
Substitution into (7.19) then gives
(7.20)
This power series is called the MacLaurin series
expansionof the functionf(x).
The MacLaurin series is an expansion of the function
f(x) about the pointx 1 = 10 (Figure 7.2); that is, the value
of the function at point xis expressed in terms of the
values of the function and its derivatives atx 1 = 10. For
this to be possible the function and its derivatives
must exist atx 1 = 10 andthroughout the interval 0 to x. In
addition, the expansion is valid only within the radius of
convergence of the series.
Examples of MacLaurin series
- The binomial series
11
The binomial series is the MacLaurin expansion of the function(1 1 + 1 x)
afor arbitrary
values of a. We have
f(x) = 1 (1 1 + 1 x)
af′(x) 1 = 1 a(1 1 + 1 a)
a− 1f′′(x) 1 = 1 a(a 1 − 1 1)(1 1 + 1 x)
a− 2f′′′(x) 1 = 1 a(a 1 − 1 1)(a 1 − 1 2)(1 1 + 1 x)
a− 3f
(n)(x) 1 = 1 a(a 1 − 1 1)(a 1 − 1 2)=(a 1 − 1 n 1 + 1 1)(1 1 + 1 x)
a−nWhenx 1 = 10 , the factor(1 1 + 1 x)
a−nis replaced by 1,
f(0) 1 = 1 1, f′(0) 1 = 1 a, f′′(0) 1 = 1 a(a 1 − 1 1), f′′′(0) 1 = 1 a(a 1 − 1 1)(a 1 − 1 2), =
and the MacLaurin expansion is
(7.21)
()() ()( )
11
1
2
12
3
23+=++
−
!
−−
!
xax +
aa
x
aa a
x
afx f
x
f
x
f
x
() ()=+() () f ()
!
′ +
!
′′ +
!
0 ′′′ +
1
0
2
0
3
0
23==
!
=∑
nnnx
n
f
00
∞()()
....................................................................................................................................................................................................................................................0 x
x
f(0)
f(x)
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Figure 7.2
11The binomial theorem (series) was discovered by Newton in 1665 and described in 1676 in letters written to
Henry Oldenburg, secretary of the Royal Society, for transmission to Leibniz. The theorem was published by
Wallis in his Algebraof 1685.