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section10notes - Lecture 10 • Definitions of...

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Unformatted text preview: Lecture 10 • Definitions of nondecreasing, nonincreasing and monotonic sequences. • Theorem: All bounded and monotone sequences converge. • Example: Show that the sequence defined by s0 = 1 and sn+1 = converges. What is its limit? √ sn + 1 • Theorem: Unbounded and montone sequences diverge to ∞ or −∞. • Define lim sup sn and lim inf sn . Note the difference between lim sup sn and sup{sn }. • Theorem: If lim sn is a real number or ±∞, then lim inf sn = lim sn = lim sup sn . • Theorem: If lim sup sn = lim inf sn , then lim sn exists and is equal to both. • Definition of a Cauchy sequence. • Theorem: A sequence is Cauchy iff it converges. 1 1 • Use the Cauchy criterion to prove that the sequence defined by sn = 1+ 2 +· · ·+ n diverges. 1 ...
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