Page 284 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 275

Thus, the approximation P 4 ð:3Þ for sin :3is correct to four decimal
places.
Additional insight to the process of approximation of functional
values results by constructing a graph of P 4 ðxÞ and comparing it to
y ¼ sin x. (See Fig. 11-2.)
x 3
P 4 ðxÞ¼ x
6
ffiffiffi
p
The roots of the equation are 0; 6. Examination of the first and Fig. 11-2
ffiffiffi
p
2 and a relative
second derivatives reveals a relative maximum at x ¼
p ffiffiffi
minimum at x ¼ 2. The graph is a local approximation of the sin
curve. The reader can show that P 6 ðxÞ produces an even better approximation.
(For an example of series approximation of an integral see the example below.)
SOME IMPORTANT POWER SERIES
The following series, convergent to the given function in the indicated intervals, are frequently
employed in practice:
3 5 7 2n 1
x x x x
1. sin x ¼ x þ þ ð 1Þ n 1 þ 1 < x < 1
3! 5! 7! ð2n 1Þ!
x 2 x 4 x 6 n 1 x 2n 2
2. cos x ¼ 1 þ þ ð 1Þ þ 1 < x < 1
2! 4! 6! ð2n 2Þ!
x 2 x 3 x n 1
x
3. e ¼ 1 þ x þ þ þ þ þ 1 < x < 1
2! 3! ðn 1Þ!
x 2 x 3 x 4 n 1 x n
2 3 4 n
4. ln j1 þ xj ¼ x þ þ ð 1Þ þ 1 < x @ 1
3 5 7 2n 1
x x x x
1 þ x
ln
5. 1 ¼ x þ þ þ þ þ þ 1 < x < 1
2 1 x 3 5 7 2n 1
3 5 7 2n 1
x x x n 1 x
1
6. tan x ¼ x þ þ ð 1Þ þ 1 @ x @ 1
3 5 7 2n 1
p pð p 1Þ 2 pð p 1Þ ... ð p n þ 1Þ n
2! n!
7. ð1 þ xÞ ¼ 1 þ px þ x þ þ x þ
This is the binomial series.
(a) If p is a positive integer or zero, the series terminates.
(b) If p > 0 but is not an integer, the series converges (absolutely) for 1 @ x @ 1:
If 1 < p < 0, the series converges for 1 < x @ 1:
ðcÞ
(d) If p @ 1, the series converges for 1 < x < 1.
For all p the series certainly converges if 1 < x < 1.
1
ð
2
x
x
EXAMPLE. Taylor’s Theorem applied to the series for e enables us to estimate the value of the integral e dx.
! 0
4

8
6
2 Ð 1 x x x e 10
1 x
2
Ð
Substituting x for x,we obtain x dx
0 e dx ¼ 0 1 þ x þ 2! þ 3! þ 4! þ 5!
where
1 4 1 6 1 8
2! 3! 4!
P 4 ðxÞ¼ 1 þ x þ x þ x þ x
and

e
10
x ; 0 < < x
5!
R 4 ðxÞ¼