APPM 1360 Midterm: appm1360fall2015exam3_sol
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Fall 2015: determine if the series is absolutely convergent, conditionally convergent or divergent, justify your answer: (a)(10 pts) ( 1)n n2/3. Solution: (a)(10 pts) note thatx|an| = ( 1)n n2/3 is an alternating series with bn = 1/n2/3 and note that bn is decreasing since (n + 1)2/3 > n2/3 implies bn+1 < bn. 1 n(ln n)2 ( 1)n+1en (n + 1)! (c)(10 pts) Xn=0 n2/3 which is a divergent p-series and note thatx an = Xn=1 n2/3 and clearly lim n bn = 0 and so the original series converges by alternating series test and thus is conditionally convergent. (b)(10 pts) we use the ratio test, note that lim n (cid:12)(cid:12)(cid:12)(cid:12) an+1 an (cid:12)(cid:12)(cid:12)(cid:12) = 0 < 1 (n + 1)! ( 1)n+1en(cid:12)(cid:12)(cid:12)(cid:12) and so is absolutely convergent by ratio test. ( 1)n+1en (n + 1)! Xn=0 (c)(10 pts) first note that the series is positive, so it is either absolutely convergent or divergent.
