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

698 Chapter 11 Infinite Sequences and Series
The Squeeze Theorem can also be adapted for sequences as follows (see Figure 7).
Squeeze Theorem for Sequences
c n
If a n < b n < c n for n > n 0 and lim
n l `
a n − lim
n l `
c n − L, then lim
n l `
b n − L.
Another useful fact about limits of sequences is given by the following theorem,
whose proof is left as Exercise 87.
b n
6 Theorem
If lim
n l `
| a n| − 0, then lim
n l ` an − 0.
a n
0 n
FIGURE 7
The sequence hb nj is squeezed
between the sequences ha nj
and hc nj.
This shows that the guess we made earlier
from Figures 1 and 2 was correct.
Example 4 Find lim
n l `
n
n 1 1 .
SOLUTION The method is similar to the one we used in Section 2.6: Divide numerator
and denominator by the highest power of n that occurs in the denominator and then use
the Limit Laws.
lim
nl `
n
n 1 1 − lim
nl `
1
1 1 1 n
− 1
1 1 0 − 1
−
lim 1
nl `
lim 1 1 lim
nl ` nl `
1
n
Here we used Equation 4 with r − 1.
Example 5 Is the sequence a n −
n
s10 1 n
convergent or divergent?
SOLUTION As in Example 4, we divide numerator and denominator by n:
n
lim
nl `
n
s10 1 n − 1
lim
nl `
Î 10
n 1 1 2 n
− `
because the numerator is constant and the denominator approaches 0. So ha n j is
divergent.
ln n
Example 6 Calculate lim
n l ` n .
SOLUTION Notice that both numerator and denominator approach infinity as n l `.
We can’t apply l’Hospital’s Rule directly because it applies not to sequences but to
functions of a real variable. However, we can apply l’Hospital’s Rule to the related
function f sxd − sln xdyx and obtain
n
ln x 1yx
lim − lim
x l ` x x l ` 1 − 0
Therefore, by Theorem 3, we have
ln n
lim
n l ` n − 0 n
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