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Unformatted text preview: Ratio and Root Tests John E. Gilbert, Heather Van Ligten, and Benni Goetz Properties of geometric series enable us to test whether other series converge or diverge  even if theyre not geometric series! Root Test: an infinite series n a n Converges Absolutely when lim n  a n  1 /n < 1 , Diverges when lim n  a n  1 /n > 1 , if lim n  a n  1 /n = 1 , the test is inconclusive, it tells us nothing. Example 1: does the series n =1 ( 1) n 1 n 2 n + 1 n converge absolutely ? Solution: the Root Test works well here because of the n thpower exponent in a n = ( 1) n 1 n 2 n + 1 n . For then  a n  1 /n = n 2 n + 1 n 1 /n = n 2 n + 1 1 2 as n . So the series converges absolutely by the Root Test. Whats the connection with geometric series? Well, lim n  a n  1 /n = L =  a n  L n for all large n, so n  a n  n L n ....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Sadler
 Geometric Series, Multivariable Calculus

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