4130final - MATH 4130 FINAL EXAM Math 4130 final exam 18...

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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 . (b) Explain what is meant by limsup n y n . (c) Show that if lim n →∞ x n = L then lim n →∞ | x n | = | L | . (d) Suppose limsup n y n = L . Is it necessarily true that limsup n | y n | = | L | ? Explain your answer. (2) (20 marks.) A real number α 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 ( α ) = 0. (In this case, we say that α is a root of f .) If α is not algebraic, it is said to be transcendental ....
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4130final - MATH 4130 FINAL EXAM Math 4130 final exam 18...

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