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 definition. Definition 2.3.2 A sequence {an}is said todiverge to∞if for everyM∈R, there exists N∈Nsuch that an >Mfor 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∈Nsuch that an <Mfor all n≥N. Remark 2.3.2 We should not confuse a sequence that diverges to∞(that is, one that satisfies the previous definition), with a divergent sequence (that is, one that does not converge). ■ Example 2.3.4 Consider the sequence {an}given by an = n2 +1 5n . We will show, using Definition 2.3.2, that limn→∞an =∞. Let M∈R. Note that n2 +1 5n = n 5 + 1 5n ≥ n 5 . Choose N>5M. Then, if n≥N, we have an ≥ n 5 ≥ N 5 >M. 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 n→∞ an =∞. Similarly, if {an}is decreasing and not bounded below, then lim n→∞ an =−∞. Proof: Fix any real number M. Since {an} is not bounded above, there exists N∈Nsuch that aN ≥M. Then an ≥aN ≥Mfor 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 n→∞ an =∞, lim n→∞ bn =∞, and lim n→∞ cn =−∞ Then
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