APPM 1360 Midterm: appm1360spring2018exam3sol
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Spring 2018: (16 pt) determine whether the series is absolutely convergent, conditionally convergent, or divergent. cos3 n. Xn=1 (a) (8 pt) by the direct comparison test, 0 (cid:12)(cid:12)(cid:12)(cid:12) (p = 3 > 1) so cos3 n (b) cos3 n n3. Xn=1 n3 (b) (8 pt) apply the ratio test. n3 (cid:12)(cid:12)(cid:12)(cid:12) )2 is convergent and therefore is absolutely convergent . The series is absolutely convergent : (22 pt) suppose you know that f (x) = (ln n)(x 6)n n has a radius of convergence of r = 1. Xn=3 (a) use this information to determine the values of x for which the series is absolutely or conditionally convergent. (b) find the value of f (17)(6). Solution: (a) (16 pt) the power series is centered at 6 and has a radius of convergence of r = 1. It is absolutely convergent on (5, 7) and possibly at the endpoints.
