Unformatted text preview: series were positive; in general we simply consider the absolute values of the terms and we end up testing for absolute convergence. THEOREM 10.39 The Ratio Test Suppose that lim n →∞  a n +1 /a n  = L . If L < 1 the series ∑ a n converges absolutely, if L > 1 the series diverges, and if L = 1 this test gives no information. Proof. The example above essentially proves the ﬁrst part of this, if we simply replace 1 / 5 by L and 1 / 2 by r . Suppose that L > 1, and pick r so that 1 < r < L . Then for n ≥ N , for some N ,  a n +1   a n  > r and  a n +1  > r  a n  ....
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 Spring '07
 JonathanRogawski
 Math, Calculus, lim, general form, Root Tests, argument work

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