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MidtermICorrection - MAT 319/320 Real Analysis Correction...

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MAT 319/320 Real Analysis Correction of Midterm I Problem 1. (25 points) Let C > 0 be a real number. Show that for any natural number n 1 , one has (1+ C ) n 1+ nC . Proof. Proof by induction: Let’s 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. a) First version:(detailed solution) For any n greaterorequalslant 1 , let’s factor by the dominant terms under the square root: x n = 1 n +1 (1+2 n )( n +3) radicalbig = 1 n +1 n.
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