MATH 1005 Study Guide - Midterm Guide: Alternating Series, Ratio Test, Horse Length
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Two linearly independent solutions of the system [cancel] are (cid:1) (cid:1), e2t(cid:0) 1 1: e 2t(cid:0) 1 4 (cid:1) (cid:1), et(cid:0) 1 1, e t(cid:0) 1 4. 4n is n=0: e 2t(cid:0) 4 1 (cid:1), e t(cid:0) 1 1 (cid:1) (cid:26) x(cid:48) = 3x + y (cid:1), e2t(cid:0) 1, e t(cid:0) 1 4 y(cid:48) = 4x + 2y. Determine whether the following series is convergent or divergent. [2] ( 1)n n2+1 is alternating series with bn = 1 n2+1 which is decreasing with limn 1 n2+1 = 0, 8 so series converges. (cid:1)n = converges according to ratio test where (cid:12)(cid:12)(cid:12)(cid:12) an+1 (cid:12)(cid:12)(cid:12)(cid:12) = lim n ((n + 1)2( ) = lim n (n + 1)2( 3. For each one of the following series, determine whether it converges absolutely, converges conditionally, or diverges. 5n which converges based on ratio test where n=1.
