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Unformatted text preview: Math 104, Practice Midterm 1
The ﬁrst midterm will cover all the following sections: §1 – 5 and §7 – 11 in Ross. Time: July 9, 2:304:00pm, in lecture room 3107 Etcheverry. It will be a closedbook, closednotes exam and no calculator will be allowed. The following are some practice problems. The real exam has different format and fewer questions. 1. Use limit theorems to evaluate the following limits. Justify each step. (a) lim (b) lim 5n2 + sin n n→∞ n2 + 1 9n2 + n − 3n
n 2 n→∞ 1 + 2(−1) (c) lim n→∞ n √ 3 2. Show that 1 + 2 is irrational. 3. Suppose limn→∞ sn = ∞ and limn→∞ tn < 0. Prove that limn→∞ (sn tn ) = −∞. 4. Prove that the sequence sin 5. True/False questions. (a) The ﬁeld Q of rational numbers satisfy the completeness axiom. (b) For a bounded sequence (sn ) in R, lim sup sn = sup{sn : n ∈ N}. (c) If r ∈ Q and x is irrational, then r + x is irrational. 6. State what it means for a sequence (sn ) to diverge to inﬁnity. (Notation: lim sn = ∞.)
n→∞ nπ 2 diverges. 1 7. Prove that if lim √ n n→∞ sn = L < 1, then
n→∞ lim sn = 0. 8. If s1 = √ 2, and sn+1 = 2+ √ sn , ∀n ≥ 1, prove that (sn ) converges. 9. Show that if every subsequence of (sn ) has a further subsequence that converges to the same s, then (sn ) converges to s. 2 ...
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 Winter '08
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 Math

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