APPM 1360 Midterm: appm1360summer2016exam3_sol
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Summer 2016: (35 points) determine whether the series is absolutely convergent, conditionally convergent, or divergent. Show all work, and state any theorems or tests that you use. (a) (b) (c) (d) (e) Solution: (a) let p an = p 1. N+4 and p bn = p 1 p-series, we conclude by the lct that p an is also divergent (b) the function f (x) = Since p bn is a divergent x(2+ln x)3/2 is continuous, positive, and decreasing for x 2. 2 dx =(cid:20) 2 x (2 + ln x)3/2. 2 + ln 2 as t . Since the corresponding improper integral converges, the series must also converge by the integral test. Since the terms are positive, the series is absolutely convergent. N 1 as n . Since the terms are all positive, the series is absolutely convergent bn = bn+1 = By the alternating series test, the series is convergent.
