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Assignment 5, Math 220 Due: Friday, March 8th, 2002 1 Use the − N definition of convergence to show that the sequence an = 3n+1 converges. 2n+5 2 Suppose that (xn ) is a sequence of real numbers converging to x, √ with xn ≥ 0 for all n ∈ N.√Show that the sequence ( xn ) of positive square roots converges to x. √ √ 3 Let x1 = 2 and xn+1 = 2 + xn for n ∈ N. Show that (xn ) converges and find the limit. 4 Establish the convergence or divergence of the sequence yn , where 1 1 1 + +···+ for n ∈ N. yn = n+1 n+2 2n 5 Give an example of an unbounded sequence with a convergent subsequence. √ 6 If xn = n, show that (xn ) satisfies lim |xn+1 − xn | = 0 but that it is not a Cauchy sequence. 7 If x1 < x2 are arbitrary real numbers and xn = 12 (xn−2 + xn−1 ) for n > 2, show that (xn ) is convergent. What is its limit? 8 By using partial fractions, show that ∞ X 1 1 = . n(n + 1)(n + 2) 4 n=1 1