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MATH 201 Section B Spring 2016 Extra Credit Problems 1. For each of the following sequences, find the limit and prove that it is correct, directly from the definition 2.1.2 of limit. 4n2 + 2 (a) xn = 2 3n − 7 √ √ (b) xn = n + 1 − n (Hint: find an equivalent expression for xn .) sin2 (n3 ) (c) xn = n √ 2. Let x1 = 1 and xn+1 = 1 + xn for n = 1, 2, . . . (This is a recursive definition for the sequence. Read Section 2.2.3 for a related example.) (a) Write out the first four terms in the sequence. (b) Show by induction that xn ≤ 2 for all n. (c) Show by induction that {xn }∞ n=1 is monotone increasing. (d) It now follows by Theorem 2.1.10 that x = limn→∞ exists. Find the value of x. (Suggestion: find an equation which x must satisfy.)