The Integral Test; p-SeriesDivergence TestConvergence CriterionSlide 4Try It Outp-SeriesSlide 7AssignmentThe Integral Test; p-SeriesLesson 8.3Divergence Test•Be careful not to confuseSequence of general terms { ak }Sequence of partial sums { Sk }•We need the distinction for the divergence testIf Then must divergelim 0kka���ka�Note this only tells us about divergence. It says nothing about convergenceNote this only tells us about divergence. It says nothing about convergenceConvergence Criterion•Given a series•If { Sk } is bounded aboveThen the series convergesOtherwise it diverges•NoteOften difficult to applyNot easy to determine { Sk } is bounded above with 0 for all k ka a k��Convergence Criterion•Integral Test•Given ak = f(k)k = 1, 2, …f is positive, continuous, decreasing for x ≥ 1•Then eitherboth converge … orboth diverge11( )kka and f x dx��=��Try It Out•Given Does it converge or diverge?Consider( )3/ 212kk�-=+�( )3/ 21lim 2bbx dx-��+�p-Series•DefinitionA series of the formWherep is a positive constantp-Series test•Converges if p > 1•Diverges if p ≤ 111 1 1 1...1 2 3p p p pkk�== + + +�11pkk�=�Try It Out•Given series•Use the p -series test to determine if it converges or diverges 1kke�-=�Assignment•Lesson 8.3•Page 520•Exercises 3 – 53
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