RedAssign12[1].1 - If the limit equals one, you cannot...

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Unit: Sequences and Series Module: The Ratio and Root Tests The Ratio Test www.thinkwell.com info@thinkwell.com Copyright 2001, Thinkwell Corp. All Rights Reserved. 1591 –rev 10/02/2001 To apply the ratio test to a series = 1 n n a , let + →∞ = 1 lim n n n a a r . If ρ < 1, then the series converges absolutely . If ρ > 1, then the series diverges. If ρ = 1, then the test is inconclusive. For a series to converge, its terms must decrease at a fast rate. You can study this by using a ratio to compare two consecutive terms, a n +1 and a n . As you take the limit of this ratio, you learn about the behavior of the series. In fact, since the ratio test involves the absolute values of the terms, you can determine absolute convergence , which is the stronger form. If the limit of the ratio is less than one, then the terms are decreasing fast enough for the series to converge absolutely. If the limit of the ratio is greater than one, then the series will diverge.
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Unformatted text preview: If the limit equals one, you cannot conclude anything. You must try a different test. You know this series converges absolutely because it is a geometric series with a base that is positive and less than one. Will the ratio test agree that the series converges? To apply the ratio test, you need to take the limit of the ratio of two consecutive terms a n +1 and a n . Dont forget the absolute value symbol! Since you have a fraction over a fraction, invert and multiply. The terms are already positive, so you can remove the absolute value symbol. Canceling 5 n leaves one in the numerator and five in the denominator. Use the Greek letter rho ( ) to represent the value of the limit. Since is less than one, the series converges absolutely. Remember, if equals one, you cannot conclude anything. Use another test. The series might converge like this p-series does, or it might diverge....
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