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MATHEMATICS: ASSIGNMENT 6 There are two parts to this assignment: Part A and Part B. Part A is to be completed online, and Part B is to be handed in, both before 10:00 a.m. on Friday, January 15. Part A This part of the assignment can be found online, labelled Assignment6A, at webwork.elearning.ubc.ca. Part B 1. (a) Let a1 = 1 and an+1 = √ 1 + 2an for n = 1, 2, 3, . . .. Show that the sequence {an } converges. 1. (b) Find the limit of the sequence in part (a). 2. Consider the sequence {an } where an = bxc + b2xc + · · · + bnxc , n2 and x is any real number. (Here “bac” is defined to be the largest integer which is at most a.) Find the limit of the sequence, if it exists. 3. Recall that the Fibonacci sequence is defined thus: F1 = F2 = 1, and Fn + Fn+1 = Fn+2 for n = 1, 2, 3, . . .. Prove (by induction, or otherwise) that F12 + F22 + · · · + Fn2 = Fn Fn+1 for n = 1, 2, 3, . . .. 4. (a) Let {an } be a sequence with lim a2n = L and lim a2n+1 = L. Does {an } converge? Justify your n→∞ n→∞ answer. √ 4. (b) Let a1 = 1 and an+1 = 1 + 1/ (1 + an ) for √ n = 1, 2, 3, . . .. Show that {an } converges to 2. (This is called the continued fraction expansion of 2; every real number has such an expansion.) 5. Prove that if X n≥1 an converges, then {an } converges to 0.