MTH 253 Calculus (Other Topics)Does the Series Converge?The Comparison TestSlide 4Using the Comparison TestExamples w/ the Comparison TestThe Limit Comparison TestSlide 8Example w/ the Limit Comparison TestMTH 253Calculus (Other Topics)Chapter 11 – Infinite Sequences and SeriesSection 11.4 – Comparison 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 Comparison TestNOTE:For all series in this section, it will be assumed that each term is non-negative.That is, given thenka�0, ka k� "The Comparison TestBasic Idea:If you can show that a first series is less than a second series, and the second series is know to be convergent, then so is the first series (sn increasing w/ upper bound).If you can show that a first series is greater than a second series, and the second series is divergent, then so is the first series (sn increasing wo/ upper bound).k ka b�� �k ka b�� �Difficult Part? Finding the series to compare.Using the Comparison Test1. Make an “educated” guess.•convergent or divergent?2. Find a series and prove your guess.•usually similar to the originalHelpful Ideas:Increasing the numerator or decreasing the denominator gives something bigger.Decreasing the numerator or increasing the denominator gives something smaller.Examples w/ the Comparison Test19 6k +�1 1 19 6 9k k k< <+Smaller than the divergent harmonic series NOTHING!42k k+�4 4 42 2 12k k k k< = �+Smaller than twice a convergent p-series Convergent21kk k+-�2 2 21 1k k kk k k k k k+> > =- -Larger than the divergent harmonic series DivergentThe Limit Comparison TestBasic Idea:If one series is essentially a constant multiple of another series (except for possibly the first finite number of terms), then they have the same behavior.lim 0 then & both converge/divergelim 0 and converges, then convergeslim and diverges, then divergesnn nnnnn nnnnn nnnaa bbab abab ab������>==�� �� �� �Difficult Part? Finding the series to compare.The Limit Comparison TestProof of part 1 (parts 2 & 3 are similar) …lim 0nnnacb��= >Implies by the definition of a limit w/ =c/2 …2nnaccb- <Therefore …2 2nnac ccb- < - <32 2n n nc cb a b< < converges convergesn nb a�� � diverges divergesn nb a�� �Example w/ the Limit Comparison Test19 6k +�1 1 19 6 9k k k< <+Smaller than the divergent harmonic series NOTHING!Comparison Test Fails!1 9 6 1lim lim1 9 6 9k kk kk k�� ��+= =+Therefore, same behavior
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