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# section9notes - Lecture 9 • Deﬁnition of a bounded...

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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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## This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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