MTH 253 Calculus (Other Topics)Does the Series Converge?The Ratio & Root TestsThe Ratio TestSlide 5Slide 6Slide 7Example w/ the Ratio TestThe Root TestExample w/ the Root TestMTH 253Calculus (Other Topics)Chapter 11 – Infinite Sequences and SeriesSection 11.5 – The Ratio and Root TestsCopyright © 2009 by Ron Wallace, all rights reserved.Does the Series Converge?1kka�=� 10 Tests for ConvergenceGeometric SeriesN-th Term Test (Divergence Test)Integral Testp-Series TestComparison TestLimit Comparison TestRatio TestRoot TestAlternating Series TestAbsolute Convergence TestEach test has it limitations (i.e. conditions where the test fails).The test tells you nothing!The Ratio & Root TestsNOTE:For all series in this section, it will be assumed that each term is positive.That is, given thenka�0, ka k> "The Ratio Test1Let: limkkkaLa+��=�1 where kkar L r k Kae+< + < " >21 1 1...kk k ka ra r a r a+ -< < <1 1 kka r a+<\� ��Geometric Series!Convergent if |r|<1 Comparison Test If L < 1, the series converges.(from the limit definition)The Ratio Test1Let: limkkkaLa+��=�1 where kkas s L k Kae+> < - " >21 1 1...kk k ka sa s a s a+ -> > >1 1 kka s a+>\� ��Geometric Series!Divergent if |s|>1 Comparison Test If L > 1, the series diverges.(from the limit definition)The Ratio Test1Let: lim 1kkkaLa+��= =If L=1, the test fails!1 divergesk�1 ( 1)lim lim 11 1k kk kk k�� ��+= =+21 convergesk�2 22 21 ( 1)lim lim 11 2 1k kk kk k k�� ��+= =+ +The Ratio Test1Let: limkkkaLa+��=Given where 0 k ka a k> "�• If L < 1, the series converges.• If L > 1, the series diverges.• If L = 1, the test fails.Example w/ the Ratio Test11 2 222 2244 4( 1)lim lim lim 4 14( 1) 4 2 1kkkkk k kk kkk k kk++�� �� ��+= � = = >+ + +24kk� Divergent!The Root TestLet: limkkka L��=Given where 0 k ka a k> "�• If L < 1, the series converges.• If L > 1, the series diverges.• If L = 1, the test fails.Proof is similar to the ratio test!Example w/ the Root Test1 1 1lim lim 12 2 2kk kkk ke e- -�� ��� �- -= = <� �� �12kke-� �-� �� ��
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