APPM 1360 Midterm: archive_appm1360summer2018exam3_sol
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Summer 2018: (21 pts) determine whether the series is absolutely convergent, conditionally convergent, or divergent. ( 1)n (2n)! Solution. (a) (7 pts) lim n (cid:12)(cid:12)(cid:12)(cid:12) an+1 an (cid:12)(cid:12)(cid:12)(cid:12) Thus, by the ratio test this series diverges. (b) (7 pts) first note that cos(n ) = ( 1)n, We have that 1 n > 0 for n 1, 1 n > 1 n+1 for. Thus, by the alternating series test this series converges conditionally. (c) (7 pts) set f (x) = xe x. We have f (x) > 0 for x 1, f (x) = (1 x)e x < 0 for x > 1, and n f (x) is continuous. The integral test is then suitable. xe xdx e xdx. Thus, since the terms in the series are all positive the series converges absolutely by the integral. Test: (18 pts) consider the sequences {an} n=1 and {bn} n=1 and that. No justi cation is necessary for the following questions.
