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

740 Chapter 11 Infinite Sequences and Series
is convergent. (Recall that a finite
number of terms doesn’t affect convergence.) Therefore o a n is absolutely convergent.
(ii) If | a n11ya n | l L . 1 or | a n11ya n | l `, then the ratio | a n11ya n | will eventually
be greater than 1; that is, there exists an integer N such that
. 1 whenever n > N
is also convergent. It follows that the series o ǹ−1 | a n |
Z
a n11
a n
Z
This means that | a n11 | . | a n | whenever n > N and so
lim
nl` an ± 0
Therefore o a n diverges by the Test for Divergence.
n
− 1, the test gives no
information. For instance, for the convergent series o 1yn 2 we have
Note Part (iii) of the Ratio Test says that if lim n l ` | a n11ya n |
Z
a n11
a n
Z −
1
sn 1 1d 2
1
n 2
−
n 2
sn 1 1d − 1
2
S1 1
nD
1 l 1 as n l `
2
whereas for the divergent series o 1yn we have
The Ratio Test is usually conclusive if
the nth term of the series contains an
exponential or a factorial, as we will
see in Examples 4 and 5.
Z
a n11
a n
Z −
1
n 1 1
1
n
−
n
n 1 1 − 1
1 1 1 n
l 1
as n l `
Therefore, if lim n l ` | a n11 ya n | − 1, the series o a n might converge or it might diverge.
In this case the Ratio Test fails and we must use some other test.
Estimating Sums
In the last three sections we used
various methods for estimating the sum
of a series—the method depended on
which test was used to prove convergence.
What about series for which the
Ratio Test works? There are two possibilities:
If the series happens to be an
alternating series, as in Example 4,
then it is best to use the methods of
Section 11.5. If the terms are all
positive, then use the special methods
explained in Exercise 46.
Example 4 Test the series ò
n−1
s21d n n3
n
for absolute convergence.
3
SOLUtion We use the Ratio Test with a n − s21d n n 3 y3 n :
Z
a n11
a n
Z −
s21d n11 sn 1 1d 3
− 1 3
S n 1 1
n
3 n11 sn 1 1d3
− ? 3n
s21d n n 3 3 n11 n 3
3 n
3
D − 1 S1 1 1 l
3 nD3 1 3 , 1
Thus, by the Ratio Test, the given series is absolutely convergent.
n
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