hw3 - (a) b n + (-1) n n B * (b) b 1 n 2 + (-1) n 3 n B *...

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MATH 410: Homework 3 Due in class Friday 2/17/2012 Section 2.2 1. For each of the following statements determine whether it is true or false and justify informally: (a) Every bounded sequence converges. * (b) A convergent sequence of positive numbers has a positive limit. * (c) The sequence { n 2 + 1 } converges. * (d) A convergent sequence of rational numbers has a rational limit. * (e) The limit of a convergent sequence in the interval ( a, b ) also belongs in ( a, b ). * Note: This is book problem 1. 2. Show that every x R is the limit of a sequence of irrational numbers. * Note: This is book problem 3. Section 2.3 1. Justify whether each of the following sequences is monotone.
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Unformatted text preview: (a) b n + (-1) n n B * (b) b 1 n 2 + (-1) n 3 n B * Note: This is book problem 2. 2. Suppose that { a n } is monotone. Prove that { a n } converges i { a 2 n } converges. Show that this ** result does not hold without the monotonicity assumption. Note: This is book problem 3. 3. (a) Use book problem 5 and the Comparison Lemma to obtain another proof (not the books) * that if | c | < 1 then lim n c n = 0. (b) Use book problem 5 and the Comparison Lemma to prove that lim n nc n = 0. ** Note: This is book problem 6. It requires (and you can assume) book problem 5....
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This note was uploaded on 02/27/2012 for the course MATH 2 taught by Professor Staff during the Spring '08 term at Maryland.

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