MATH 1B Final: MATH 1B Harvard 2004Spring1bFinal sol
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Full credit may not be given for an answer alone. You may use the backs of the pages or the extra pages for scratch work. Students who, for whatever reason, submit work not their own will ordinarily be required to withdraw from the college. 1: (15 points) test the following for convergence or divergence. This is a p-series with p = 17 > 1. So the series converges. n2 en (ii) p n=1. The ratio of successive terms is (n+1)2 en+1 n2 en. =(cid:18) n + 1 n (cid:19)2 1 e . 1 e so the series converges by the ratio test. To employ the alternating series test, we must check three things: (iii) p n=1( 1)n sin(cid:0) 1 n(cid:1) whether (cid:8)sin(cid:0) 1 n(cid:1)(cid:9) is positive, decreasing, and tends to zero. The rst is clear because sin x is positive when x is in between 0 and includes the set of all 1 checked by di erentiation:
