homework3mat25

homework3mat25 - 1 and | x n-x n +1 | k | x n-1-x n | for...

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Homework 3 due Thursday August 21st 1. Prove the following using only the definition of convergence or divergence: (a) (8.2(e)) sin( n ) n 0. (b) 4 n 2 - 3 5 n 2 - 2 n 4 5 . (c) x n = 1 + ( - 1) n does not converge to any x R . (d) y n = n ! diverges to . 2. (8.4) Let ( t n ) be a bounded sequence and let ( s n ) be a sequence that converges to 0. Prove that s n t n 0 using only the definition of convergence (no theorems). 3. Prove or give a counterexample: if ( s n ) and ( t n ) are divergent sequences, then ( s n + t n ) diverges. 4. (10.8) Let ( s n ) be a nondecreasing sequence of positive numbers and define σ n = s 1 + s 2 + ··· + s n n . Prove that σ n is also nondecreasing. 5. Let ( s n ) be the sequence defined by s 0 = 1 and s n +1 = 1 4 (2 s n + 5). Prove that ( s n ) is bounded and monotone, and find its limit. 6. A sequence ( s n ) is said to be contractive if k so that 0 < k <
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Unformatted text preview: 1 and | x n-x n +1 | k | x n-1-x n | for all n N . In this exercise I will guide you through a proof that all contractive sequences converge. (a) Prove by induction the sequence will satisfy the inequality | x n-x n +1 | k n-1 | x 1-x 2 | for all n 1. Then use this to rewrite | x n-x m | for arbitrary n,m in terms of | x 1-x 2 | . (b) (9.18(a)) Prove 1 + k + k 2 + + k n = 1-k n +1 1-k by induction. (c) Prove that k n 0. (d) Combine parts (b) and (c) to prove that &gt; 0, N so that m &gt; n &gt; N , k n + k n +1 + + k m &lt; . (e) Combine (a)-(d) to prove that all contractive sequences are Cauchy, hence convergent. 1...
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