Comparison TestsDirect Comparison TestTry It on TheseLimit Comparison TestSlide 5Example of Limit ComparisonAssignmentComparison TestsLesson 8.4Direct Comparison Test•Given •If converges, then converges •What if•What could you conclude about these?0 for all for some k ka c k N N� � �1kkc�=�1kka�=�0 for all k Nk kd a� � �1 andkkd�=�1kka�=�Try It on These•Test for convergence, divergence•Make comparisons with a geometric series or p-series10.5kk�=�/ 202kk�=�Limit Comparison Test•Given ak > 0 and bk > 0 for all sufficiently large k … and …where L is finite and positive•Then either both converge … or both diverge1kkkaLb�==�k and bka� �Limit Comparison TestStrategy for evaluating 1. Find series with know convergence and general term "essentially same"2. Verify that this limit exists and is positive3. Now you know that behaves the same as ka�kb�limknkab��ka�kb�Example of Limit Comparison•Convergent or divergent?•Find a p-series which is similar•Consider•Now apply the comparison( )111kk k�=� +�11kk�=�Assignment•Lesson 9.4•Page 628•Exercises 1 - 27
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