APPM 1360 Midterm: appm1360summer2014exam3_sol
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Summer 2014: (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. 2n ln(n) (a) looks like the series with an = 1 n4 , which is a convergent p-series. But our series is greater than a convergent series, so the direct comparison test does not work. First, note that our series terms and those of x 1 n 1 (n2 + 1)1/2 n5 n (n2 + 1)1/2 (n2+1)1/2. = lim n lim n n4 are always postive, so the lct applies. = lim n n 1 (n2 + 1)1/2 n3. 1 1 n4 ((n2 + 1) 1 n2 )1/2 n4 n5 n. Because the terms are all positive, x|an| = x an, and regular convergence is the same as absolute convergence. Therefore absolutely convergent by limit comparison test (b) this is a geometric series in disguise:
