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Name__________________________________ Calculus Period ________ Lesson 9.5: Alternating Series, Absolute Convergence and Conditional Convergence Theorem 8: The Alternating Series Test (Leibniz’s Theorem) The series 1 n 1 n 1 un u1 u2 u3 u4 ... converges if all three of the following are satisfied. 1. The un ' s are all positive. 2. un un 1 for all n 3. lim un 0 n Absolute Convergence A series an converges absolutely if the corresponding series of absolute values an converges. A series that converges, but does not converge absolutely, converges conditionally. Determine whether the series converges absolutely, converges conditionally or diverges. Give reasons for your answer. Find a bound for the truncation error after 99 terms. 1 1 1 1 1 1. 1 ... 2. 1 2 3 4 ... 3. 2 1 ... 2 3 4 2 4 4. 1 n 1 n 1 1 n 5. 1 n 1 n 1 1 n2 Radius of Convergence To determine to what values each power series converge, we use one of three tests: Geometric, Root, or Ratio. When using the Root or Ratio test, you must check the endpoints individually because the test is inconclusive at 1. In Exercises 35-50, find (a) the interval of convergence of the series. For what values of x does the series converge (b) absolutely, (c) conditionally. Also, identify the Radius of Convergence. 36. x 5 n n 0 38. n 0 3x 2 n n 44. nx n n 2 n 0 4 n 1 HW p523: 23-25, 35-43 odd