James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

768 Chapter 11 Infinite Sequences and Series
Table 1
Important Maclaurin
Series and Their Radii
of Convergence
1
1 2 x − ò x n − 1 1 x 1 x 2 1 x 3 1 ∙ ∙ ∙ R − 1
n−0
e x − ò
n−0
x n
n! − 1 1 x 1! 1 x 2
2! 1 x 3
3! 1 ∙ ∙ ∙ R − `
sin x − òs21d n
n−0
cos x − òs21d n
n−0
tan 21 x − òs21d n
n−0
x 2n11
s2n 1 1d! − x 2 x 3
x 2n
s2nd! − 1 2 x 2
x 2n11
2n 1 1 − x 2 x 3
3! 1 x 5
5! 2 x 7
2! 1 x 4
4! 2 x 6
3 1 x 5
5 2 x 7
7! 1 ∙ ∙ ∙ R − `
6! 1 ∙ ∙ ∙ R − `
7 1 ∙ ∙ ∙ R − 1
lns1 1 xd − òs21d x n
n21
n−1 n − x 2 x 2
2 1 x 3
3 2 x 4
4 1 ∙ ∙ ∙ R − 1
s1 1 xd k −
n−0S ò
nDx k n − 1 1 kx 1
ksk 2 1d ksk 2 1dsk 2 2d
x 2 1 x 3 1 ∙ ∙ ∙ R − 1
2!
3!
Example 10 Find the sum of the series
1
1 ? 2 2 1
2 ? 2 1 1
2 3 ? 2 2 1
3 4 ? 2 1 ∙ ∙ ∙.
4
SOLUTION With sigma notation we can write the given series as
òs21d n21
n−1
1
n ? 2 n
− òs21d n21 (1 2) n
n−1 n
Then from Table 1 we see that this series matches the entry for lns1 1 xd with x − 1 2 . So
òs21d n21
n−1
1
n ? 2 n − lns1 1 1 2d − ln 3 2 n
TEC Module 11.10/11.11 enables you
to see how successive Taylor polynomials
approach the original function.
One reason that Taylor series are important is that they enable us to integrate functions
that we couldn’t previously handle. In fact, in the introduction to this chapter we mentioned
that Newton often integrated functions by first expressing them as power series and then
integrating the series term by term. The function f sxd − e 2x 2 can’t be integrated by techniques
discussed so far because its antiderivative is not an elementary function (see Section
7.5). In the following example we use Newton’s idea to integrate this function.
ExamplE 11
(a) Evaluate y e 2x 2 dx as an infinite series.
(b) Evaluate y 1 0 e2x 2 dx correct to within an error of 0.001.
SOLUTION
(a) First we find the Maclaurin series for f sxd − e 2x 2 . Although it’s possible to use the
direct method, let’s find it simply by replacing x with 2x 2 in the series for e x given in
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