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L21 Absolute Convergence Ratio and Root Tests Part I

# L21 Absolute Convergence Ratio and Root Tests Part I - 1 If...

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Lecture 21: Absolute Convergence and Ratio and Root Tests Consider the two convergent alternating series (a) X ( ± 1) n +1 n and (b) X ( ± 1) n +1 n 2 Consider the series whose terms are the absolute values of the terms of the original series in (a) and (b): X 1 n and X 1 n 2 Since X 1 n diverges and the orginal series (a) converges, we say that series (a) is condition- ally convergent. Since X 1 n 2 converges, we say that series (b) is absolutely convergent. 1

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Basically the question is: Does P a n converges or diverges? Given X a n , consider the series X j a n j : Def . If X j a n j converges, then we call X a n absolutely convergent. If X j a n j diverges, but X a n converges, then we call X a n conditionaly convergent Theorem: If a series X a n is absolutely convergent, then it is convergent. 2
ex. X cos n n 2 3

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Ratio Test- Let X a n be a series.

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Unformatted text preview: 1. If lim n !1 j a n +1 a n j = L < 1 ; then the series is absolutely convergent. 2. If lim n !1 j a n +1 a n j = L > 1 ; then the series is divergent. 3. If lim n !1 j a n +1 a n j = 1 ; then the Ratio Test is inconclusive. ex. 1 X n =1 3 n n ! 4 ex. 1 X n =1 2 n n 2 ex. 1 X n =1 1 n 3 5 NYTI: 1 X n =1 ( ± 1) n +1 n 2 2 n n ! ex. 1 X n =1 n 2 n Tonight’s homework: Practice Ratio test and work out problems to test for absolute or conditional convergence or divergence. Pay attention to which ones can’t be determined by applying ratio test . What do you do to de-termine the convergence of these problems? 6...
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L21 Absolute Convergence Ratio and Root Tests Part I - 1 If...

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