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

Section 11.5 Alternating Series 735
Estimating Sums
A partial sum s n of any convergent series can be used as an approximation to the total
sum s, but this is not of much use unless we can estimate the accuracy of the approximation.
The error involved in using s < s n is the remainder R n − s 2 s n . The next theorem
says that for series that satisfy the conditions of the Alternating Series Test, the size of the
error is smaller than b n11 , which is the absolute value of the first neglected term.
You can see geometrically why the
Alternating Series Estimation Theorem
is true by looking at Figure 1 (on
page 733). Notice that s 2 s 4 , b 5,
| s 2 | s5 , b6, and so on. Notice also
that s lies between any two consecutive
partial sums.
Alternating Series Estimation Theorem If s − o s21d n21 b n , where b n . 0, is
the sum of an alter nating series that satisfies
then
(i) b n11 < b n and (ii) lim
n l ` bn − 0
| R n | − | s 2 s n | < b n11
Proof We know from the proof of the Alternating Series Test that s lies between
any two consecutive partial sums s n and s n11 . (There we showed that s is larger than
all the even partial sums. A similar argument shows that s is smaller than all the odd
sums.) It follows that
| s 2 s n | < | s n11 2 s n | − b n11 n
By definition, 0! − 1.
s21d
Example 4 Find the sum of the series ò
n
correct to three decimal places.
n−0 n!
SOLUtion We first observe that the series is convergent by the Alternating Series Test
because
1
(i)
sn 1 1d! − 1
n! sn 1 1d , 1 n!
(ii) 0 , 1 n! , 1 n l 0 so 1 n! l 0 as n l `
To get a feel for how many terms we need to use in our approximation, let’s write out
the first few terms of the series:
s − 1 0! 2 1 1! 1 1 2! 2 1 3! 1 1 4! 2 1 5! 1 1 6! 2 1 7! 1 ∙ ∙ ∙
− 1 2 1 1 1 2 2 1 6 1 1
24 2 1
120 1 1
720 2 1
5040 1 ∙ ∙ ∙
Notice that b 7 − 1
5040 , 1
5000 − 0.0002
In Section 11.10 we will prove that
e x −o ǹ−0 x n yn! for all x, so what we
have obtained in Example 4 is actually
an approximation to the number e 21 .
and s 6 − 1 2 1 1 1 2 2 1 6 1 1
24 2 1
120 1 1
720 < 0.368056
By the Alternating Series Estimation Theorem we know that
| s 2 s 6 | < b 7 , 0.0002
This error of less than 0.0002 does not affect the third decimal place, so we have
s < 0.368 correct to three decimal places.
n
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