MATH 301 Homework 5 - f inf f x n +1 ; x n +2 ; :::: g : n...

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Math 301, Homework 5 1. Give the answer with short explanations. The sets below are all considered in R . i. one lower bound if they exist. ii. iii. the set? (a) & 0 ; p 2 ± \ Q : (b) ² y 2 R : y = x 2 + 2 x 3 for some x 2 ( 1 ; 3) ³ : (c) n n n 2 +1 : n 2 Z o : (d) ² ( 1) n + 1 n : n 2 Z ³ : (e) f e n : n 2 N g : 2. Let x n = 8 < : 1 1 n if n = 3 k 1 n 2 if n = 3 k + 1 1 + 1 n if n = 3 k + 2 where k = 1 ; 2 ; 3 ; :::: (a) Find all cluster points of ( x n ) : (b) Looking at the cluster points tell what lim sup x n and lim inf x n must be. (c) Use the expressions in Proposition 1.5.6 and step by step compute inf f sup f x n +1 ; x n +2 ; :::: g : n = 1 ; 2 ; ::: g and sup
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Unformatted text preview: f inf f x n +1 ; x n +2 ; :::: g : n = 1 ; 2 ; ::: g : 3. Prove the &quot;Alternating Series Test&quot;; .ie suppose that ( x n ) is a positive decreasing sequence with lim x n = 0 : Show that the alternating series 1 X n =0 ( &amp; 1) n x n is convergent and the sum satis&amp;es j s &amp; s k j ± s k where s k is the partial sum and s = lim s k : Hint: Start by showing that ( s 2 k ) and ( s 2 k +1 ) are both monotone. 4. For n 2 N let A n = &amp; 1 n ; 1 + 1 n ´ = ² x 2 R : 1 n ± x &lt; 1 + 1 n ³ : Express the sets A = [ 1 n =1 A n ; B = \ 1 n =1 A n as intervals I A ; I B . Prove your answer by showing that I A ² A and A ² I A etc. 1...
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