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Math+301+Second+Exam+2005

# Math+301+Second+Exam+2005 - space is[1 2(with the usual...

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Math 301 Second Exam Monday, Nov. 9, 2005 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. Let X be a metric space with metric d(x,y). a) State the definition for a sequence of points in X to be a Cauchy sequence. b) State the definition for X to be a complete metric space. c) Give with proof an example of a metric space that is not complete. (Don’t forget to define the metric!) 2. Let X be a metric space with metric d(x,y). a) State the definition of a contraction on X. . b) For F(X) a function defined on define the function R(x) used in Newton’s method. (Your answer should be a function of x, not x n .) c) Let F(x) = x 2 - 2. Write the formula for R(x). BE SURE TO SIMPLIFY YOUR ANSWER AS MUCH AS POSSIBLE. (There should be two terms.) d) Prove or disprove the following statement: The mapping R from part c satisfies the hypotheses of the CMP if the metric

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Unformatted text preview: space is [1, 2] (with the usual Euclidean metric). YOU NEED NOT PROVE THAT ℝ IS COMPLETE. 3. Let X be a metric space with metric d(x,y). a) State the CMP. (Don’t forget any hypotheses or conclusions!) b) Prove the uniqueness part of the CMP. c) Prove convergence of the sequence. d) Prove the limit point is a fixed point. 4. a) Suppose S is a finite set with a metric d and suppose A is a nonempty subset of S. Prove A is closed. b). Let F(x) = xsin(1/x) when x ≠ 0 and let F(0) = 0. Prove or disprove that F is uniformly continuous on [-1,1]. c) Show there is a proper subset of ℚ with ∂A = ∅ . (Here ℚ is the metric space, with the usual Euclidean metric.) 5. a) Find the radius of convergence of the power series ∑ (n!) 3 (2n!)-1 z 2n (Notice the power of z is 2n.) b) Suppose the power series ∑ a n z n has radius of convergence R > 0. Prove that if 0 < r < R, there is a real number K such that |z| ≤ r ⇒ ∑ |a n z n | ≤ K....
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Math+301+Second+Exam+2005 - space is[1 2(with the usual...

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