5128_Ch08_434-471

440 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
EXAMPLE 9 Determining Convergence or Divergence
Determine whether the sequence with given nth term converges or diverges. If it converges,
find its limit.
(a) a n (1) n n 1
, n 1, 2, … (b) b 1 4, b n b n1 2 for all n 2
n
SOLUTION
(a) This is the sequence of Example 6 with graph shown in Figure 8.1. This sequence
diverges. In fact we can see that the terms with n even approach 1 while the terms with
n odd approach 1.
(b) This is the sequence of Example 7 with graph shown in Figure 8.3. This sequence
also diverges. In fact we can say that lim b n . Now try Exercise 35.
n→
An important theorem that can be rewritten for sequences is the Sandwich Theorem
from Chapter 2.
THEOREM 2
The Sandwich Theorem for Sequences
If lim a n lim c n L and if there is an integer N for which a n b n c n for all
n→ n→
n N, then lim b n L.
n→
EXAMPLE 10
Using the Sandwich Theorem
s n
Show that the sequence co
n
converges, and find its limit.
SOLUTION
Because cos x 1 for all x, it follows that
cos n
cos n
1
n
n n
for all integers n 1. Thus,
1 cos n 1
.
n n n
Then, lim co s n
0 because lim
n→ n
n→ 1 n n→ lim 1 n
0 and the sequence
co s n
n
converges. Now try Exercise 41.
We can use the Sandwich Theorem to prove the following theorem.
THEOREM 3
Absolute Value Theorem
Consider the sequence {a n }. If lim
n→
a n 0, then lim
n→
a n 0.
Proof We know that a n a n a n . Thus, lim a n 0 and lim a n 0 implies
n→ n→
that lim a n 0 because of the Sandwich Theorem.
■
n→
Another way to state the Absolute Value Theorem is that if the absolute value sequence
converges to 0, then the original sequence also converges to 0.