Page 283 - Schaum's Outline of Theory and Problems of Advanced Calculus

274 INFINITE SERIES [CHAP. 11

1 nþ1
f ðnþ1Þ (Lagrange form)
ðn þ 1Þ!
R n ðxÞ¼ ð Þðx cÞ ð13Þ
( is between c and x.)
(The theorem with this remainder is a mean value theorem. Also, it is called Taylor’s formula.)
For each n there exists such that
1 n
f ðnþ1Þ (Cauchy form)
n!
R n ðxÞ¼ ð Þðx Þ ðx cÞ ð14Þ
1 ð x n
ðx tÞ f ðnþ1Þ ðtÞ dt (Integral form)
R n ðxÞ¼ ð15Þ
n! c
If all the derivatives of f exist, then
1
X 1 n
f ðnÞ
n!
f ðxÞ¼ ðcÞðx cÞ ð16Þ
n¼0
This infinite series is called a Taylor series, although when c ¼ 0, it can also be referred to as a
MacLaurin series or expansion.
One might be tempted to believe that if all derivatives of f ðxÞ exist at x ¼ c, the expansion (16) would
be valid. This, however, is not necessarily the case, for although one can then formally obtain the series
on the right of (16), the resulting series may not converge to f ðxÞ. For an example of this see Problem
11.108.
Precise conditions under which the series converges to f ðxÞ are best obtained by means of the theory
of functions of a complex variable. See Chapter 16.
The determination of values of functions at desired arguments is conveniently approached through
Taylor polynomials.

EXAMPLE. The value of sin x may be determined geometrically for 0; , and an infinite number of other
6
arguments. To obtain values for other real number arguments, a Taylor series may be expanded about any of
these points. For example, let c ¼ 0 and evaluate several derivatives there, i.e., f ð0Þ¼ sin 0 ¼ 0; f ð0Þ¼ cos 0 ¼ 1,
0
v
1v
f ð0Þ¼ sin 0 ¼ 0; f ð0Þ¼ cos 0 ¼ 1; f ð0Þ¼ sin 0 ¼ 0; f ð0Þ¼ cos 0 ¼ 1.
00
000
Thus, the MacLaurin expansion to five terms is
1 1
3 5
3! 51
sin x ¼ 0 þ x 0 x þ 0 x þ
Since the fourth term is 0 the Taylor polynomials P 3 and P 4 are equal, i.e.,
x 3
P 3 ðxÞ¼ P 4 ðxÞ¼ x
3!
and the Lagrange remainder is
1 5
cos x
5!
R 4 ðxÞ¼
Suppose an approximation of the value of sin :3is required. Then
1 3
P 4 ð:3Þ¼ :3 ð:3Þ :2945:
6
The accuracy of this approximation can be determined from examination of the remainder. In
particular, (remember j cos j 1)

1 243
1
5 <:000021
5! 120 10
jR 4 j¼ cos ð:3Þ 5