MAT 21C Midterm: Math 21C Midterm 1 Spring 2018 1
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MAT 21C Full Course Notes
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N(cid:19) (b) an = cos(cid:18) 1 (c) an = 1 + ( 1)n (d) an = n! nn. = lim n p1 + 2/n2: (b) since cos x is a continuous function, we have lim n cos(cid:18) 1. N(cid:19) = cos 0 = 1: (c) the sequence diverges since its terms oscillate between 0 (for n odd) and 2 (for n even), (d) we have an = 1 2 3 . n n n n . n. Since 0 an 1/n and 1/n 0, we have an 0 by the sandwich theorem. 1: [10pts] find the sums of the following series. (a) (b) 3n (cid:18) n 1 n n n + 1(cid:19) Solution: (a) this is a geometric series with ratio r = 2/3. Since |r| < 1, the series converges and the sum is. 5: (b) this is a telescoping series of the form. Xn=2 (bn bn+1) , bn = n 1 n.
