48 2.3 Monotone Sequences When a monotone sequence is not bounded, it does not converge. However, the behavior follows a clear pattern. To make this precise we provide the following defnition. Defnition 2.3.2 A sequence {an} is said to diverge to ∞ if for every M ∈ R, there exists N ∈ N such that an > M for all n ≥ N. In this case, we write limn →∞ an = ∞. Similarly, we say that {an} diverges to −∞ and write limn →∞ an = −∞ if for every M ∈ R, there exists N ∈ N such that an < M for all n ≥ N. Remark 2.3.2 We should not confuse a sequence that diverges to ∞ (that is, one that satisfes the previous defnition), with a divergent sequence (that is, one that does not converge). ■ Example 2.3.4 Consider the sequence {an} given by 2n + 1 an = . 5n We will show, using Defnition 2.3.2, that limn →∞ an = ∞. Let M ∈ R. Note that 2n + 1 n 1 n = + ≥ . 5n 5 5n 5 Choose N > 5M. Then, if n ≥ N, we have n N an ≥ ≥ > M. 5 5 The following result completes the description of the behavior of monotone sequences. Theorem 2.3.3 If a sequence {an} is increasing and not bounded above, then lim an = ∞. n→∞ Similarly, if {an} is decreasing and not bounded below, then lim an = −∞. n→∞ Proof: Fix any real number M. Since {an} is not bounded above, there exists N ∈ N such that aN ≥ M. Then an ≥ aN ≥ M for all n ≥ N because {an} is increasing. Therefore, limn →∞ an = ∞. The proof for the second case is similar. □ Theorem 2.3.4 Let {an}, {bn}, and {cn} be sequences of real numbers and let k be a constant. Suppose lim an = ∞, lim bn = ∞, and lim cn = −∞ n→∞ n→∞ n→∞ Then
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