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Unformatted text preview: Tuesday, February 7 − Lecture 18: Tests for convergence of series : Comparison test. (Refers to Section 8.4 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Apply the comparison test to determine convergence or divergence of a series, show a reasonable error bound for the value of a series known to converge by the integral test or by the fact it is a geometric series. 18.1 Theorem − The Comparison test . Let be two series such that 0 ≤ a i ≤ b i for all i . Then the following statements hold true : Proof: Part I : In this part the Monotone convergence (sequence) theorem is invoked. Given: Since 0 ≤ a i ≤ b i for all i . So the sequences { A n } and { B n } are such that A n ≤ B n for all n . We are also given that the sequence { B n } converges, say to L . Required to show: That the sequence { A n } must converge to some number. Part II : Given: The sequences { A n } and { B n } are such that A n ≤ B n for all n and that the...
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 Winter '08
 ZHOU
 Calculus, Mathematical Series

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