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Student number Name [SURNAME(S), Givenname(s)] MATH 101, Section 212 (CSP) Week 8: Marked Homework Assignment Due: Thu 2011 Mar 10 14:00 HOMEWORK SUBMITTED LATE WILL NOT BE MARKED 1. Find the limit for each the following sequences, or else determine that the sequence is divergent. (a) limn→∞ 2n+1 √ 1−3 n (b) limn→∞ n+(−1)n+1 n (c) limn→∞ [1 + (−1)n ] sin n n n! limn→∞ 106n (d) limn→∞ (e) 2. Let a1 is given, and let an+1 = 4an (1 − an ) for n = 1, 2, 3, . . .. Find all values of a1 such that {an } is a constant sequence (every an is the same value). √ 3. (a) If a is positive and a < 2, prove that a < 2a. √ (b) If a is positive and a < 2, prove that 2a < 2. √ √ (c) Let a1 = 2 and let an+1 = 2an for n = 1, 2, 3, . . .. Prove that the sequence converges. (d) Find limn→∞ an . 4. For each of the following series, determine if it is convergent or divergent. If it is convergent, find the sum. (a) P∞ (b) P∞ (c) n=0 1 2n + (−1)n 5n 1 n=1 cos n P∞ 6n+3 n=1 n2 (n+1)2 5. Express 1.24123 = 1.24123123123 . . . as a ratio of integers. 6. For each of the following series, find the values of x for which the series converges, and find the sum of the series for those values of x. (a) (b) P∞ n=1 P∞ n=0 2 n xn sinn x