This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonnegative 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....
View
Full
Document
This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters

Click to edit the document details