Penn Math Math 202: Proving Things: Analysis Fall 2013

Faculty: Jerry L. Kazdan
    Telephone: (215) 898-5109
    email: kazdan AT math.upenn.edu
    Office Hours: Wed. 10:30-11:30   (and also by appointment) in DRL 4E15
TA: Haomin Wen
    Telephone: (215) 573-5043
    email: weh AT math.upenn.edu
    Office Hours: Mon 2:30-3:30 & Thurs 6:30-7:30 (and also by appointment) in DRL 3N2D
    Web page: http://www.math.upenn.edu/~weh/

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication.
We will study real and complex numbers, sequences, series, continuity, differentiability and integrability for functions of one variable, proving our way as we go, and enjoying a number of challenging problems.
This course is about thinking, attempting to understand. It it for students who enjoy thinking hard, even when completely stumped. Then understanding (when it comes) is all the sweeter.

Intuition and computational skill will be essential in the discovery and presentation of your ideas.

The concept of a proof will be vital. There is nothing exotic about a proof. It is simply convincing someone else about your reasoning. You give different proofs to different people, depending on their background — and how well you know them. Thw first and most important person to convince is yourself.
Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening. The evening labs will provide opportunity for all students to present their solutions at the blackboard, and to become comfortable and proficient at doing this.
To be successful in this course, you should be present for all class meetings and plan to take good notes.

Text: John P. D'Angelo and Douglas B. West Mathematical Thinking: Problem-Solving and Proofs, Second edition, Prentice Hall (2000). [ href="http://www.math.uiuc.edu/~west/mt/mt2err.html"> Typos, and corrections.]

Course Grading

Prerequisites & Review Material: Some experience with calculus.
To remove rust from your background I suggest doing the problems from recent Math 103 Final Exams. Note that both Math 103 and 104 focus on techniques for solving standard calculus problems. This course will pay more attention to the basic ideas and proofs.

Problems to think about during August.
On the first day of class some students might be invited to present their solutions at the blackboard.

Some References

Notes:
Symmetries of a Square
Axioms for a Field
Real solution of x2 = 2
Infinite Series in the Complex Plane These are Lecture Notes I wrote years ago. Chapter 1 discusses Infinite Series allowing complex numbers. To me, allowing complex numbers makes power series easier.
Newton's method for square roots.
e is an irrational number.
Class notes from Oct. 29
sin x + sin2x +...+sin nx = ?
TeX/LaTeX Stuff Creating documents having formulas.
Appliction to differential equations This remarkable application uses most of the ideas in Math 202. It is a good review. The almost identical version, Appliction to differential equations* has a slightly stronger result -- but is a bit more fussy.
How do you prove that?
Solution to HW 11 Bonus Problem 1

Homework Assignments:

Exams: There will be three in-class exams, from 12:00-1:20. You may always use one 3"×5" card with handwritten notes on both sides