Transcription of A GUIDE TO THE LIMIT COMPARISON TEST
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A GUIDE TO THE LIMIT COMPARISON TEST
A GUIDE TO THELIMITCOMPARISONTESTJ oseph BreenThe LIMIT COMPARISON test is the GOAT infinite series convergence test, but knowing when and how to useit effectively can be difficult. This GUIDE explains the intuition, subtleties, and heuristics of the test andhopefully provides enough elucidating examples. The blue links below are hyperlinks to each The statement of the LIMIT COMPARISON test12 When and how to use LIMIT quick summary .. I: Isolating obvious dominant behavior .. II: Pesky logarithms .. III: Trig functions and other oddities (Taylor polynomials) .. IV: A last ditch resort ..63 Some examples71 The statement of the LIMIT COMPARISON testIn order to use LIMIT COMPARISON , we have to know the statement. I ll provide the mathematical statement,but also how you shouldthinkabout the ( LIMIT COMPARISON test.).Let n=1anbe an infinite series withan>0. Letbn>0be apositive sequence.(i)Iflimn anbnis a finite, positive number, then n=1anand n=1bneither both converge ordiverge.
1 The statement of the limit comparison test In order to use limit comparison, we have to know the statement. I’ll provide the mathematical statement, but also how you should think about the statement. Theorem 1.1 (Limit comparison test.). Let P 1 n=1 a n be an infinite series with a n > 0. Let b n > 0 be a positive sequence. (i) If lim n!1 ...
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