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. Defne 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 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 7→ 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 ,n =0 , 1 , 2 ,... . ±ind 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 ±rst ( n = 0) is equal to 0. Therefore the G.L.B. is 0. The second term ( n =1 )i s 1 2 , after which the sequence is decreasing. Therefore the L.U.B. is 1 2 . 1
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4. Let { a n } be a sequence of real numbers with lim n a n = L . Suppose that a 0 =0 and a n a n 1 µ 1 3 n , for all n =1 , 2 , 3 ,....
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This note was uploaded on 04/23/2008 for the course MATH 142A taught by Professor Fitzsimmons during the Fall '01 term at UCSD.

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

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