hw7 - = 1-1 n x n and let s k be the sequence of partial...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 7 Due date: Oct 15 (Wed) Exercises from the textbook. 3.5: 2(a); 3(b); 9 3.7: 2; 3(b)–(c); 5; 11; 13 (HINT for 13: a n a n + 1 max { a n ,a n + 1 } 2 ( a n + a n + 1 ) 2 .) Out-of-textbook exercises. 1. Let ( x n ) be a Cauchy sequence and let ( x n k ) k = 1 be a subsequence. Show directly that if ( x n k ) k = 1 x then ( x n ) n = 1 x . You may not use the fact that Cauchy sequences converge. 2. Let ( x n ) be a decreasing sequence of non-negative reals. Consider the alternating series n =
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Unformatted text preview: = 1 (-1 ) n x n and let ( s k ) be the sequence of partial sums, i.e. s k = ∑ k n = 1 (-1 ) n x n . (a) Show that the subsequence ( s 2 k-1 ) ∞ k = 1 is increasing, while the subsequence ( s 2 k ) ∞ k = 1 is decreasing. Conclude that they converge (possibly to different limits). (b) Use part (a) to prove that the series ∑ ∞ n = 1 (-1 ) n x n converges if and only if lim n →∞ x n = 0....
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