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?
- More useful Wikipedia links:
- Wikipedia does not have the unit step function as I defined it. Here is its closest relative: the Heaviside function.
- Rectangular function
- In several definitions and proofs we used dichotomy of real numbers.
- In several definitions and proofs we used trichotomy of real numbers.
- But, in some proofs dichotomy and trichotomy were not sufficient. Then we used
quadruplicity. Thank you Timothy for The Oxford English Dictionary reference.
- Tuesday, March 31, 2009
-