Unformatted text preview: f inf f x n +1 ; x n +2 ; :::: g : n = 1 ; 2 ; ::: g : 3. Prove the "Alternating Series Test"; .ie suppose that ( x n ) is a positive decreasing sequence with lim x n = 0 : Show that the alternating series 1 X n =0 ( & 1) n x n is convergent and the sum satis&es j s & 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 = & 1 n ; 1 + 1 n ´ = ² x 2 R : 1 n ± x < 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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 Fall '08
 ALBERTERKIP
 Math, Sets, 3k, lim sup xn, lim inf xn, Let xn

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