# Spring 2009 MATH 226: Limits and infinite series Branko Ćurgus

Wednesday, June 10, 2009

• This is what I wrote during the exam.

Monday, June 1, 2009

Friday, May 29, 2009

• This is what I wrote during the exam.

Thursday, May 21, 2009

Tuesday, May 19, 2009

Thursday, May 14, 2009

Monday, May 11, 2009

• This is an extended version of Axioms for the Real Numbers. In this file I prove that our version of the Completeness Axiom (which is in fact taken from the book Mathematical analysis by Vladimir Zorich, translated by Roger Cooke, published by Springer in 2004) is equivalent to the standard Completeness Axiom. As I mentioned in class, I prefer Zorich's version of the Completeness Axiom since it is stated using only the concepts introduced in preceding axioms. Moreover there is some symmetry in this axiom which is lacking in the standard version.

Monday, May 4, 2009

Wednesday, April 29, 2009

• I am posting an extended version of the section New limits from old. In class we did only one proof. Here I include all the proofs with all the details. I encourage you to read this extended version and I welcome your comments.

Tuesday, April 28, 2009

• This is what I wrote during the exam. Most of the time I spent trying to do Problem 1a without using a calculator. One way to do this is to consider the function f(x) = e^x - x^e for x > 0. As an interesting calculus problem one can prove that the function f has a unique global minimum at x=e. As a consequence, one gets f(pi) > f(e) = 0. Hence |e^pi - pi^e| = e^pi - pi^e is not an exception. For all x > 0 we have |e^x - x^e| = e^x - x^e. It is interesting that e, the famous e, is the only number with this property. That is, for each a > 0 such that a \neq e the function g(x) = a^x - x^a (x > 0) has negative values. If you find simple proofs of these statements using calculus I will give you extra credit.

• I am posting Megan's, Leslie's and Tim's solutions for Exam 1. They are all excellent.

Monday, April 27, 2009

Wednesday, April 22, 2009

• By a popular demand I am posting some hints for the second part of Assignment 1.

Tuesday, April 21, 2009

Saturday, April 18, 2009

Friday, April 17, 2009

Thursday, April 16, 2009

Monday, April 13, 2009

• Here I collected 12 different ways to say: If p, then q.

• This might be helpful for limit proofs.

Friday, April 10, 2009

• The first part of Assignment 1

• Do you need more graph paper?