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Unformatted text preview: MAT 319/320 Real Analysis Correction of Midterm I Problem 1. (25 points) Let C > be a real number. Show that for any natural number n 1 , one has (1 + C ) n 1 + nC . Proof. Proof by induction: Lets call P ( n ) the following proposition: (1 + C ) n 1 + nC . 1. P (1) is true because 1 + C greaterorequalslant 1 + C ; 2. Assume that P ( n ) is true: thus we assume that (1 + C ) n 1 + nC . Now one has (1 + C ) n +1 = (1 + C ) . (1 + C ) n greaterorequalslant (1 + C ) . (1 + nC )( because P ( n ) is true ) = 1 + nC + C + nC 2 greaterorequalslant 1 + ( n + 1) C Therefore P ( n + 1) is true. Conclusion: we proved by induction that the result is true for any integer n greaterorequalslant 1 . square Problem 2. (25 points) First version: Find lim ( x n ) , where x n = 1 n + 1 (1 + 2 n )( n + 3) radicalbig . Second version: Find lim ( x n ) , where x n = 1 n + 1 ( n + 2)(3 n + 1) radicalbig . Proof....
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This note was uploaded on 01/31/2011 for the course AMS 319 taught by Professor Mark during the Fall '10 term at SUNY Stony Brook.
- Fall '10