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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.
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• Use the Cauchy criterion to prove that the sequence deﬁned by sn = 1+ 2 +· · ·+ n
diverges. 1 ...
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 Winter '11
 Wiley
 Differential Equations, Equations

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