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

700 Chapter 11 Infinite Sequences and Series
Notice that the expression in parentheses is at most 1 because the numerator is less
than (or equal to) the denominator. So
0 , a n < 1 n
We know that 1yn l 0 as n l `. Therefore a n l 0 as n l ` by the Squeeze Theorem.
n
Example 11 For what values of r is the sequence hr n j convergent?
SOLUTION We know from Section 2.6 and the graphs of the exponential functions in
Section 1.4 that lim x l ` a x − ` for a . 1 and lim x l ` a x − 0 for 0 , a , 1. Therefore,
putting a − r and using Theorem 3, we have
It is obvious that
If 21 , r , 0, then 0 , | r |
lim r −H`
n
n l ` 0
if r . 1
if 0 , r , 1
lim
n l ` 1n − 1 and lim 0 n − 0
n l `
, 1, so
lim | r n | − lim | r | n − 0
n l ` n l `
and therefore lim n l ` r n − 0 by Theorem 6. If r < 21, then hr n j diverges as in
Example 7. Figure 11 shows the graphs for various values of r. (The case r − 21 is
shown in Figure 8.)
r=1
0
n
_1<r<0
r>1
a n
1
1
a n
1
1
FIGURE 11
The sequence a n − r n
0 n
0<r<1
r<_1
n
The results of Example 11 are summarized for future use as follows.
9 The sequence hr n j is convergent if 21 , r < 1 and divergent for all other
values of r.
lim −H r 0 if 21 , r , 1
n
n l ` 1 if r − 1
10 Definition A sequence ha n j is called increasing if a n , a n11 for all n > 1,
that is, a 1 , a 2 , a 3 , ∙ ∙ ∙ . It is called decreasing if a n . a n11 for all n > 1.
A sequence is monotonic if it is either increasing or decreasing.
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