Calculus 2nd Edition Rogawski

SECTION 11.7 Taylor Series 601
EXAMPLE 5 Substitution Find the Maclaurin series for e −x2 .
Solution Substitute −x 2 in the Maclaurin series for e x .
e −x2 =
∞∑
n=0
(−x 2 n
)
=
n!
∞∑ (−1) n x 2n
n=0
n!
= 1 − x 2 + x4
2! − x6
3! + x8
4! − ··· 2
The Taylor expansion of e x is valid for all x, so this expansion is also valid for all x.
EXAMPLE 6 Integration Find the Maclaurin series for f (x) = ln(1 + x).
Solution We integrate the geometric series with common ratio −x (valid for |x| < 1):
1
1 + x = 1 − x + x2 − x 3 +···
∫
dx
ln(1 + x) =
1 + x = x − x2
2 + x3
3 − x4
∞ 4 +···= ∑
n=1
n−1 xn
(−1)
The constant of integration on the right is zero because ln(1 + x) = 0 for x = 0. This
expansion is valid for |x| < 1. It also holds for x = 1 (see Exercise 84).
In many cases, there is no convenient general formula for the Taylor coefficients, but
we can still compute as many coefficients as desired.
n
1
y
EXAMPLE 7 Multiplying Taylor Series Write out the terms up to degree five in the
Maclaurin series for f (x) = e x cos x.
Solution We multiply the fifth-order Taylor polynomials of e x and cos x together, dropping
the terms of degree greater than 5:
)
(1 + x +
)(1 x2
2 + x3
6 + x4
24 + x5
− x2
120 2 + x4
24
Distributing the term on the left (and ignoring terms of degree greater than 5), we obtain
)
)( (1 + x + x2
2 + x3
6 + x4
24 + x5
−
(1 + x + x2
120
2 + x3 x
2
) ( x
4
)
+ (1 + x)
6 2
24
= 1 + x − x3
3 − x4
6 − x5
} {{
30
}
Retain terms of degree ≤ 5
x
321
We conclude that the fifth Maclaurin polynomial for f (x) = e x cos x is
−1
FIGURE 1 Graph of T 12 (x) for the power
series expansion of the antiderivative
∫ x
F (x) = sin(t 2 )dt
0
T 5 (x) = 1 + x − x3
3 − x4
6 − x5
30
In the next example, we express the definite integral of sin(x 2 ) as an infinite series.
This is useful because the integral cannot be evaluated explicitly. Figure 1 shows the graph
of the Taylor polynomial T 12 (x) of the Taylor series expansion of the antiderivative.
EXAMPLE 8 Let J =
∫ 1
0
sin(x 2 )dx.
(a) Express J as an infinite series.
(b) Determine J to within an error less than 10 −4 .