a3 - Department of Mathematics and Statistics McGill...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Department of Mathematics and Statistics McGill University MATH 242 Assignment 3 Due in class, Friday October 23 Full credit can be obtained from correct answers to questions totalling 70. 1. (10 points) Let a > 0 and suppose that x 3 1 > a . We define x n +1 = 2 x 3 n + a 3 x 2 n and verify that x n +1 > 0 inductively. (i) Show that x 3 n +1 - a = (8 x 3 n + a )( x 3 n - a ) 2 27 x 6 n and x n +1 - x n = - x 3 n - a 3 x 2 n . (ii) Show that x 3 n > a for all n N and that ( x n ) is a decreasing sequence bounded below by zero. (iii) Deduce that ( x n ) converges to some real number x 0 . Show that x 3 = a . 2. (10 points) Prove or disprove by counterexample the following statement. If I j are open intervals for j N and Q [ j =1 I j , then [ j =1 I j = R . 3. (10 points) Let ( x n ) n =1 be a sequence of real numbers. If x kn + ` converges as n → ∞ for all integers k and ` with k 2 and ` 1 , show that x n converges. 4. (10 points)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/05/2009 for the course MATH MATH 242 taught by Professor Drury during the Fall '09 term at McGill.

Page1 / 2

a3 - Department of Mathematics and Statistics McGill...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online