Math 1B: Calculus Project : The Abel and Dirichlet Tests Instructor: Alexander Paulin Student Name and ID : Whilst we have a large selection of series tests, there are still examples of innite series that fall outside of their reach. In this project we'll look at two powerful new tests. The Abel Test If {an} and {b,,,} are two sequences, then 00 DO 0 E an convergent 1 E nub", convergent R: 0 {bn} bounded and monotonic ":1 The Dirichlet Test If {an} and {bu} are two sequences, then 0 {an} decreasing DO ' {351300-11 = 0 :> Z nub-n convergent I The sequence of partial sulns Of{b\"}is hounded "=1 Project : The Abel and Dirichlet Tests Co 2. Prove that the following series is convergent. n You may use the following fact: Hint : Look for a . 2 sin( ) cos(a) = sin(a + ) - sin(a - ;). telescoping sum Carefully justify your answer. Solution