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 k1 ) ∞ 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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 Spring '08
 JUNGE
 Math, Mathematical analysis, Limit of a sequence

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