APPM 1360 Midterm: appm1360summer2013exam3_sol
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Summer 2013: (30 pts) determine whether the following series are conditionally convergent, absolutely convergent, or divergent. You may not use the ratio or root test for this problem. 1 p n 1 is continuous for all x, positive for x > 0, and decreasing eventually: (a) integral test: f (x) = 2xe x2. Now since the terms are all positive, x|an| = x an, which we just found is convergent. Absolutely convergent by integral test. e udu = lim t (e 1 e t2. , so this series is alternating and has bn = So use the ast: (b) first, note that cos(n ) n . Therefore the series is at least convergent, by alternating series test. But we must check if it is absolutely convergent or just conditionally convergent. This is a divergent p-series (p = 1). Xn=1 (cid:12)(cid:12)(cid:12)(cid:12) original series is conditionally convergent. (c)
