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Unformatted text preview: Lecture 9
• Deﬁnition of a bounded sequence.
• Theorem: If (sn ) converges, it is bounded.
• Give a counterexample to the statement: If (sn ) is bounded, then it converges.
• Theorem: If sn → s and tn → t, then sn + tn → s + t.
• Theorem: If sn → s and tn → t, then sn tn → st.
• Theorem: If sn → s and sn = 0 ∀n and s = 0, then 1
sn → 1.
s • Corollary: If sn → s and tn → t, sn = 0 ∀n, and s = 0, then tn
sn t
→ s. • Example: Use the above theorems to ﬁnd
n3 + 6 n2 + 7
.
n→∞ 4n3 + 3n − 4
lim • Make sure to read 9.7(a)(d), and be able to reproduce the proofs.
• Deﬁnition of a sequence diverging to ±∞. What is the diﬀerence between this
and merely being unbounded?
• Prove n2 − 3n diverges to ∞.
• Prove n2 +3
n+1 diverges to ∞. • Theorem: If (sn ) diverges to ∞ and lim tn exists and is positive, then (sn tn )
diverges to ∞.
• Theorem: (sn ) diverges to ∞ iﬀ 1
sn → 0. • Theorem: If ∃M so that ∀n ≥ M sn ≤ tn , and if sn → s and tn → t, then s ≤ t. 1 ...
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 Winter '11
 Wiley
 Differential Equations, Equations

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