APPM 1360 Midterm: appm1360spring2016exam3_sol
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Spring 2016: (24 pts, 8 pts each) determine whether the following series are absolutely convergent, conditionally convergent, or divergent. Explain your work and name any tests that you use. (a) 1 . 4 3 4 = is a convergent p-series with p = 4/3 > 1. Xn=1 (b) the series diverges by the test for divergence: lim n (cid:18)1 + Note that the root test evaluates to 1, which is inconclusive. (c) note that a comparison with a p-series will not work here. This series is less than all divergent p-series and greater than all convergent p-series. First observe that the function creasing for x 3. Z b (cid:20) 5 x(ln x)3 dx = lim b z b. 5 x(ln x)3 is positive, continuous and de- Since the integral converges, the series because the terms are all positive. 5 n(ln n)3 converges and therefore is absolutely convergent. Alternate solution: for the improper integral, note thatz .
