# hw5 - Modern Analysis Homework 5 Due 2010 The following...

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Unformatted text preview: Modern Analysis, Homework 5 Due ?, 2010 The following problems are optional , and are worth 5 pts each: E1. [30] Let P ( x ) be a degree n polynomial with integral coefficients and no rational root. Suppose that z is a root of P . Prove that there exists A > 0 such that for all rationals p/q ∈ Q we have z- p q > 1 A | q | n . Suggestion: First try for n = 2. Hint: Suppose not, then consider the integers q n P ( p q ). E2. (a)[25] Use problem E1 to construct an explicit transcendental number. (b)[22] Use problem E1 to construct uncountably many explicit 1 transcendental numbers. E3. [22] Suppose that { a n } is a sequence of non-negative reals. Prove that ∑ a n converges if and only if Q (1+ a n ) converges. Note: Feel free to assume that e x = ∑ ∞ k =0 x n n ! for all x ∈ R . E4. [20] Rudin, Ch 3, problem 7: prove that the convergence of ∑ a n implies the convergence of X √ a n n , if a n ≥ 0. Note: This is an “aha!” problem which can be solved with a dirty trick....
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hw5 - Modern Analysis Homework 5 Due 2010 The following...

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