The Root Test:The Ratio Test THEOREM 12.4.1 THE ROOT TEST Leta be a series with nonnegative terms,and suppose that (a)k→p. ()）Ifp1,then ag diverges. (iii)If p =1,then the test is inconclusive.The series may converge;it may diverge. Main Meny
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Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Root Test; The Ratio Test Example ( ) 1 ln k k
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The Root Test;The Ratio Test THEOREM 12.4.2 THE RATIO TEST Let ag be a series with positive terms and suppose that ak+1→入 ak ()If入l,then∑ak diverges. (iii)If=1,then the test is inconclusive.The series may converge;it may diverge. Main Meny o
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The Root Test;The Ratio Test Summary on Convergence Tests In general,the root test is used only if powers are involved.The ratio test is particularly effective with factorials and with combinations of powers and factorials.If the terms are rational functions of k,the ratio test is inconclusive and the root test is difficult to apply.Series with rational terms are most easily handled by limit comparison with a p-series,a series of the form E 1/kP.If the terms have the configuration of a derivative,you may be able to apply the integral test.Finally,keep in mind that,if ag0,then there is no reason to try any convergence test;the series diverges. Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Root Test; The Ratio Test Summary on Convergence Tests In general, the root test is used only if powers are involved. The ratio test is particularly effective with factorials and with combinations of powers and factorials. If the terms are rational functions of k, the ratio test is inconclusive and the root test is difficult to apply. Series with rational terms are most easily handled by limit comparison with a p-series, a series of the form Σ 1/kp . If the terms have the configuration of a derivative, you may be able to apply the integral test. Finally, keep in mind that, if ak 0, then there is no reason to try any convergence test; the series diverges. →