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Unformatted text preview: R satisfying  a n  6 b nb n +1 for all n ∈ N , prove that ∑ a k converges absolutely. Proof : We will use again the Cauchy Criterion; let p 6 q be arbitrary positive integers. Then q X k = p  a k  6 b pb q +1 , by hypothesis. Since { b n } is monotone decreasing and convergent, hence Cauchy, for arbitrary ε > 0 there exists some n ∈ N such that for all q > p > n we have  b pb q +1  = b pb q +1 < ε. Thus, for all q > p > n , we have q X k = p  a k  < ε 1 thus ∑ a k converges absolutely by the Cauchy Criterion. Remark Note that the hypothesis lim b n = 0 was not used! We only used the fact that b n is convergent. 2...
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 Winter '08
 hitrik
 Convergence, Natural number, Dominated convergence theorem, ak bk

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