Why Read This Book? - Index of

EXERCISE 6.1.15 Consider the sequence 〈an〉 defined by
�
10, if n is odd
an =
1 − n2 , if n is even
Prove that 〈an〉 is bounded from above but not from below.
6.1 Sequences Defined 189
To close out this introductory section on sequences, here is an example that
illustrates a technique we will use in Section 6.2. The exercises that follow will be
similar.
Example 6.1.16 Consider the sequence defined by an = (n + 3)/(n − 1) for
n ≥ 2.
(a) Find a positive integer N such that |aN − 1| < 0.001.
(b) Does the inequality in part (a) hold for all n ≥ N?
Solution
(a) Expanding the inequality |aN − 1| < 0.001 according to Theorem 2.3.4
and then solving for N yield the following equivalent inequality statements.
−0.001 <
N + 3
− 1 < 0.001
N − 1
−0.001 < 4
< 0.001
N − 1
−0.001(N − 1) −3999 and N>4001
The inequality N>4001 is stronger than N>−3999, so we may let N = 4002.
(b) Suppose n ≥ N. Then
−0.001 < 0 < 4 4
≤ < 0.001 (6.6)
n − 1 N − 1
so that |an − 1| < 0.001 for all n ≥ N. �
EXERCISE 6.1.17 Let 〈an〉 be defined as in Example 6.1.16, and let ɛ>0be
given. Find a value of N (which will be in terms of ɛ) such that |aN − 1|