MATH 3150 Midterm: MATH 3150 Midterm Exam Spring 2011

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Instructions: write your name in the space provided. Calculators are permitted, but no notes are allowed. Each problem is worth 10 points : let a1 = 0, a2 = 1, a3 = 3, . , an = 1 + 2an 1. (a) show that the sequence {an} is strictly increasing. (b) show that the sequence {an} is bounded above. (c) show that the sequence {an} is converging. Give a reason for your answer. (d) find lim n an. Spring 2011: let {xn} be a sequence in a complete metric (x, d). (a) suppose d(xn+1, xn) 1/2n, for all n 1. Show that {xn} converges. (b) suppose d(xn+1, xn) 1/n, for all n 1. Show by example that {xn} may not converge. Spring 2011: let (x, d) be a metric space, a a subset of x, and x a point in x. Spring 2011: decide whether each of the following series converges or not.