lec13 - Theorem 13.3 Let t n be a sequence of real numbers...

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Math 117 – May 17, 2011 13 Convergence criteria for sequences Monotone Sequences Definition 13.1. A sequence of real numbers ( s n ) is called increasing , if s n +1 s n for all n N . The sequence is called decreasing , if s n +1 s n for all n N . A sequence is called monotone , if it is decreasing or increasing. A sequence would be called strictly increasing/decreasing, if the inequalities in the definition are strict. The first criteria for convergence of a sequence uses monotonicity and boundedness. Theorem 13.2 (Monotone convergence criterion) . Let ( s n ) be a sequence of real numbers. ( a ) If ( s n ) is increasing and bounded above, then ( s n ) converges. ( b ) If ( s n ) is decreasing and bounded below, then ( s n ) converges. It is not hard to see what the behavior of a monotone unbounded sequence is.
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Unformatted text preview: Theorem 13.3. Let ( t n ) be a sequence of real numbers. ( a ) If ( t n ) is increasing and unbounded above, then lim s n = + ∞ . ( b ) If ( t n ) is decreasing and unbounded below, then lim t n =-∞ . Cauchy Sequences Definition 13.4. A sequence of real numbers ( s n ) is said to be a Cauchy sequence , if for any ± > 0, there exists N , such that m,n > N implies | s n-s m | < ± . Before stating the Cauchy convergence criterion, we state two useful lemmas. Lemma 13.5. Every convergent sequence is Cauchy. Lemma 13.6. Every Cauchy sequence is bounded. Theorem 13.7 (Cauchy convergence criterion) . A sequence of real numbers is convergent iff it is Cauchy....
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