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Lecture 21: Ratio and Root Tests (10.5) Recall X Def. If |an| converges, then we call X an absolutely convergent. X X |an| diverges, but an converges, then X we call an conditionaly convergent. If We have many examples of absolutely convergent series, every convergent series in the Integral test and the comparison test lectures is an absolutely convergent series. We saw an alternating series can convergent, when the non-alternating version does not. This kind of series converge conditionally. However, if a series is absolutely convergent, then it converges, this is a stronger convergence. 1 X Theorem: If a series an is absolutely convergent, then it is convergent. ex. Does ∞ X cos n n=1 n2 converge absolutely? Explain. Ratio TestX Let an be a series and suppose that an+1 , then L = lim n→∞ an 1. If L < 1, the series converges absolutely, 2. If L > 1, the series diverges, 3. If L = 1, the test is inconclusive. 2 Notice that the Ratio test gives us more than just convergence. If the ratio test shows a series converges, then it converges absolutely. The ratio test is most effective on series that contain a lot of multiplication. And if there is an exponential or factorial term in your series, the ratio test is a good first try to determine the convergence. (However, Ratio test does NOT work some series including ’p−series’ like series–see NYTI example 3.). Determine if the series converges absolutely or conditionally: 1. ∞ X 3n n=1 n! 3 2. ∞ X 2n n=1 n2 1·3 1·3·5 n−1 1 · 3 · 5 · · · (2n − 1) 3. 1− + +· · ·+(−1) + 3! 5! (2n − 1)! ··· 4 NYTI: Do the series converge absolutely? Explain. 1. ∞ X (−1) 2 n n+1 n 2 n=1 2. n! ∞ X (−1)nn n=1 2n 3. Show that both convergence and divergence ∞ X are possible when L = 1 by considering n2 and ∞ X n=1 n−2 and ∞ X n=3 n=1 ln n . n When you work on homework problems, pay attention to which ones can’t be determined by applying ratio test. What do you do to determine the convergence of these problems then? 5