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Fall 2001 - Fitzsimmons' Class - Final Exam

Fall 2001 - Fitzsimmons' Class - Final Exam - Math 142A(P...

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Math 142A (P. Fitzsimmons) Final Exam December 7, 2001 Solutions 1. Define of each of the following terms (or statements): (a) Cauchy sequence (of real numbers) (b) least upper bound (of a bounded set S of real numbers) (c) radius of convergence (of a power series n =0 a n x n ) (d) inverse function (of a one-to-one function f ) (e) f is continuous at the real number x 0 2. Evaluate the indicated limits. (a) lim n + n 4 n . lim x 0 + x 2 cos(1 /x ) . Solution. (a) Notice that n 2 n for all n 1. This is clear for n = 1; proceeding inductively, we have n + 1 2 n + 1 2 n + 2 n = 2 n +1 . Therefore, 0 n 4 n 1 2 n , which tends to 0 as n → ∞ by a theorem established in class. The limit in question is therefore 0 by the Squeeze Theorem. (b) Because | x 2 cos(1 /x ) | ≤ x 2 and x x 2 is continuous, the Squeeze Theorem applies, and we conclude that the limit in question is 0. 3. Consider the sequence a n = n n 2 + 1 ,n = 0 , 1 , 2 ,... . Find the least upper bound and the greatest lower bound of { a n } , if they exist. Solution. Every term of the sequence is 0 and the first ( n = 0) is equal to 0. Therefore the G.L.B. is 0. The second term ( n = 1) is 1 2 , after which the sequence is decreasing. Therefore the L.U.B. is 1 2 .
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