For each of the following series state whether the series is

For each of the following series, state whether the series is absolutely convergent, conditionally convergent, or divergent. Credit will no be given unless the reasons for your conclusions are explicitly stated!

Solution

a and b are conditionally convergent by the alternating series tests, dince the terms decrease and approach zero as n gets large. They are not absolutely convergent. a) diverges by the integral test since its antiderivative 2 ln |2K+1| does not approach zero as n gets large. b) diverges by the comparion test with 1/n, which diverges and is smaller. c) converges absolutely by the ratio test. If we use the ratio test we get k/(2k+2)(2k+1). Which goes to zero as k gets large.