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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 th-power 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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