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Limit Comparison Test
LIMIT COMPARISON TEST Let and be positive term series and c a positive real number. Suppose that a_{K} ≤ cb_{K} for all infinite K. Then:
PROOF Assume converges. Let H and K be infinite. By the Cauchy Convergence Test (Section 9.2). b_{H+1} + b_{H+2} + ... + b_{K} ≈ 0. Hence 0 ≤ a_{H+1} + ... + a_{K }≤ cb_{H+1} + ... + cb_{K} = c(b_{H+1} + ... + b_{K}) ≈ 0. It follows that a_{H+1} + ... + a_{K} ≈ 0 and converges.


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