n_13489 - (p.735(i If lim n →∞ n q | a n | = L< 1...

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§ 11.6 Absolute convergence and the ratio and root tests 1.Theorem3 (p.732) If a series a n is absolutely convergent, then it is convergent. Remark: By using this result, we can generalize the use of Integral Test and Comparison Test by determining the convergence of | a n | . 2.The Ratio Test (p.733) (i) If lim n →∞ fl fl fl a n +1 a n fl fl fl = L < 1, then the series n =1 a n is absolutely convergent (and therefore convergent). (ii) If lim n →∞ fl fl fl a n +1 a n fl fl fl = L > 1, then the series n =1 a n is divergent. Remark: If lim n →∞ | a n +1 /a n | = 1, then the Ratio Test gives no information. The series a n could converge or diverge. In this case the Ratio Test fails and we must use some other test. 3.The Root Test (p.735)
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Unformatted text preview: (p.735) (i) If lim n →∞ n q | a n | = L < 1, then the series ∑ ∞ n =1 a n is absolutely convergent (and therefore convergent). (ii) If lim n →∞ n q | a n | = L > 1, then the series ∑ ∞ n =1 a n is divergent. Remark: If lim n →∞ n q | a n | = 1, then the Root Test gives no information. The series ∑ a n could converge or diverge. (If L = 1 in the Root Test, don’t try the Root Test because L will again be 1.) 1 11.6.14 ∞ X n =1 (-1) n arctan n n 3 2 11.6.16 ∞ X n =1 (-1) n +1 n 2 2 n n ! 3 11.6.24 ∞ X n =1 (-1) n n ln n 4 11.6.14 ∞ X n =1 (-1) n (arctan n ) n 5...
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