MATH+301+First+Exam+2008

# MATH+301+First+Exam+2008 - Math 301 First Exam Thursday,...

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Math 301 First Exam Thursday, Oct. 23, 2008 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy sequence of real numbers. b) Suppose z   ∈ ℂ , z 1, and suppose m,n with m < n. Write (but do not prove!) a formula (a ratio of two terms) for z m + z m+1 + ………. . + z n . c) Prove or disprove the following statement: If for all n, x n   , ∈ ℝ         and x n - x n - 1   0, then the sequence { x n } is convergent. d) Suppose {a n } is a sequence of complex numbers and |a n+1 - a n | λ n | a 1 - a 0 | , where 0 < λ < 1. Prove that the sequence is a Cauchy sequence. (Hint: You may assume as known that for any real number C, C λ n 0.) 2. a) State the definition of lim sup for a sequence {a n } of real numbers. b) State the Bolzano - Weierstrass Theorem.

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## This note was uploaded on 10/07/2009 for the course MATH 301 at Yale.

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MATH+301+First+Exam+2008 - Math 301 First Exam Thursday,...

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