RedAssign12[1].2 - Will the root test agree Take the n th...

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View Full Document Right Arrow Icon [email protected] Copyright © 2009, Thinkwell Corp. All Rights Reserved. The Root Test To apply the n th root test to a series = 1 n n a , let →∞ = lim n n n a ± . If ρ < 1, then the series converges absolutely . If ρ > 1, then the series diverges. If ρ = 1, then the test is inconclusive. The nth root test is another way to see if the terms of a series approach zero fast enough for the series to converge. To apply the root test, evaluate the limit of the n th root of the absolute value of the general term a n . Call the result rho ( ρ ). Like the ratio test, a result less that one indicates that the series converges absolutely . A result greater than one means that the series diverges. If the limit is one, then the test is inconclusive and you must try another test. This is a geometric series that you know converges.
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Unformatted text preview: Will the root test agree? Take the n th root of the absolute value of the general term a n . In the denominator, the n th root of 2 n is 2, so the limit is 1/2. Since 1/2 is less than one, the root test also shows that this series converges absolutely. In this example you have an alternating series. It looks like the influential terms are in the denominator, so this series probably converges. Since the top and bottom are both raised to the n th power, it is a good idea to try the n th root test. The absolute value excludes the alternating (–1) n . The n th root cancels the n th power in the denominator, leaving just ( n + 1). The limit is zero, so the series converges absolutely by the n th root test....
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This note was uploaded on 04/27/2010 for the course MATH 172 taught by Professor Hyon during the Spring '10 term at Community College of Denver.

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