MTH 253 Calculus (Other Topics)The Limit of Partial SumsDivergence TestGeometric Series TestP-Series TestIntegral TestComparison TestLimit Comparison TestRatio TestRoot TestAlternating Series TestAbsolute Convergence TestAbsolute ConvergenceAbsolute DivergenceConditional ConvergenceMTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesTests for Convergence – Virtual Flash CardsCopyright © 2008 by Ron Wallace, all rights reserved.1kka�=�The Limit of Partial SumsThe series converges if … … and diverges otherwise.1 21 nn k nks a a a a== = + + +�ggglimnns L��= ���1kka�=�Divergence TestThe series diverges if … … otherwise, the test fails.lim 0kka���1kka�=�Geometric Series TestThe series converges if |r| < 1 … and diverges otherwise.If it converges, then 0 kkar�=�If the series is of the form …0 1kkaarr�==-�1kka�=�P-Series TestThe series converges if p > 1 … and diverges otherwise.11 pkk�=�If the series is of the form …1kka�=�Integral TestThe integral and series have the same behavior.If the integral converges, so does the series.If the integral diverges, so does the series. ( ) when kf x a x k= = 1( )f x dx��1kka�=�Comparison TestIf …… and …… converges, then they both converge.If …… and …… diverges, then they both diverge.1 1 k kk ka b� �� ��� �1kkb���1 1 k kk ka b� �� ��� �1kkb���1kka�=�Limit Comparison TestGiven a series …If …… then both series have the same behavior.Either they both converge or they both diverge.lim 0 and finitekkkab��>1kkb���1kka�=�Ratio TestIf …… then …If L < 1 the series converges.If L > 1 the series diverges.If L = 1 the test fails.1limkkkaLa+��=1kka�=�Root TestIf …… then …If L < 1 the series converges.If L > 1 the series diverges.If L = 1 the test fails.limkkka L��=1kka�=�Alternating Series TestIf … … then the series converges, otherwise it diverges.0 ( 1) where 0 kk kka a k�=- > "�For a series of the form …1k ka a+>lim 0kka��=and1kka�=�Absolute Convergence TestIf converges, … then so does the series. Otherwise the test fails.0 kka�=�1kka�=�Absolute ConvergenceIf converges, … then converges absolutely.0 kka�=�0 kka�=�1kka�=�Absolute DivergenceIf diverges, … then diverges absolutely.0 kka�=�0 kka�=�1kka�=�Conditional ConvergenceIf diverges and converges, … then converges conditionally.0 kka�=�0 kka�=�0
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