Unformatted text preview:

MTH 253 Calculus (Other Topics)Does the Series Converge?The Comparison TestUsing the Comparison TestExamples w/ the Comparison TestThe Limit Comparison TestExample w/ the Limit Comparison TestThe Ratio TestSlide 9Slide 10Slide 11Example w/ the Ratio TestThe Root TestExample w/ the Root TestMTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.5 – The Comparison, Ratio, and Root TestsCopyright © 2006 by Ron Wallace, all rights reserved.Does the Series Converge?1kku�=�8 Tests for ConvergenceDivergence TestIntegral Testp-Series TestComparison TestLimit Comparison TestRatio TestRoot TestAlternating Series TestEach test has it limitations (i.e. conditions where the test fails).The test tells you nothing!10.410.5The 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 two series essentially differ by a constant (except for possibly the first finite number of terms), then they have the same behavior.lim 0 and finitekkkab��>Difficult Part? Finding the series to compare.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 6lim lim 91 9 6k kk kk k�� ��+= =+Therefore, same behavior  Divergent!The Ratio Test1Let: limkkkuLu+��=�1 where kkur r L k Kue+< > + " >21 1 1...kk k ku ru r u r u+ -< < <1 1 kku r u+<\� ��Geometric Series!Convergent if |r|<1 Comparison Test  If L < 1, the series converges.The Ratio Test1Let: limkkkuLu+��=�1 where kkus s L k Kue+> < + " >21 1 1...kk k ku su s u s u+ -> > >1 1 kku s u+>\� ��Geometric Series!Divergent if |s|>1 Comparison Test  If L > 1, the series diverges.The Ratio Test1Let: lim 1kkkuLu+��= =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: limkkkuLu+��=Given where 0 k ku u 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: limkkku L��=Given where 0 k ku u 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-� �-� �� ��


View Full Document

BMCC MTH 253 - Infinite Series

Download Infinite Series
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Infinite Series and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Infinite Series 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?