4130finalsols - MATH 4130 FINAL EXAM Math 4130 final exam,...

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Unformatted text preview: MATH 4130 FINAL EXAM Math 4130 final exam, 18 May 2010. The exam starts at 7:00 pm and you have 150 minutes. No textbooks or calculators may be used during the exam. This exam is printed on both sides of the paper. Good luck! (1) (20 marks.) Let { x n } and { y n } be sequences of real numbers. Let L R . (a) Explain what it means to say lim n x n = L . It means that for all > there exists N N such that if n > N then | x n- L | < . (b) Explain what is meant by limsup n y n . One definition: limsup n y n is the supremum of the set of limit-points (limits of subsequences) of the sequence { y n } . Another definition: limsup n y n is the limit of the sequence { sup k n y k } as n . (c) Show that if lim n x n = L then lim n | x n | = | L | . Suppose lim n x n = L . Let > . Then there exists N N such that if n > N then | x n- L | < . If n > N then || x n | - | L || | x n- L | < . (d) Suppose limsup n y n = L . Is it necessarily true that limsup n | y n | = | L | ? Explain your answer. No. For example, take the sequence ,- 1 , ,- 1 ,... for { y n } . Then limsup n y n = but limsup n | y n | = 1 . (2) (20 marks) A real number is said to be algebraic if for some n N there is a polynomial f ( x ) = x n + a n- 1 x n- 1 + + a of degree n with a i Q for all i , and with f ( ) = 0. (In this case, we say that is a root of f .) If is not algebraic, it is said to be transcendental . (a) Show that the set of all algebraic numbers is countable. (You may use without proof the fact that a polynomial of degree n has at most n roots.) 1 The set of all algebraic numbers is the union...
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).

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4130finalsols - MATH 4130 FINAL EXAM Math 4130 final exam,...

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