Math 347 Sec. D1 Exam, Spring 2009

Tuesday 12 May, 7:00-10:00pm, worth 30%. No books, notes, written materials, or electronic devices.

Office hours Monday 11 May, 4:30-5:30pm in Altgeld 376; Tuesday 12 May 3-4pm; or make an appointment by email.

Material The whole course, and all homework assignments.

How to study

• First make summary notes of the important definitions, theorems, ideas and methods from each section. (This effort helps you mentally organize the material.)
• Memorize the following proofs:
  • Proposition 1.4 (AGM inequality) - and why is it called the "arithmetic-geometric mean inequality" (see class notes on Corollary 1.5)
  • Proposition 3.16 (n < qⁿ by induction)
  • Proposition 4.44 (NxN is countable)
  • Theorem 5.13 (the factorial formula for "n choose k")
  • Theorem 5.17 (Binomial Theorem)
  • Theorem 5.23
  • Propositions 6.6 and 6.7 (learn the statements as well as the proofs)
  • Theorem 7.16
  • Lemma 7.19
  • Definition 7.21 and the justification that this definition make sense
  • Theorem 7.30 (Chinese Remainder Theorem). Note: the first paragraph of the proof in the text proves uniqueness of solutions modulo N, and the second paragraph proves existence of a solution.
  • Theorem 8.16 (Rational Zeros Theorem)
  • Definition 13.10 (limit/convergence of a sequence)
  • Examples 13.11 as covered in class, including that lim(a_n +b_n)=lim a_n + lim b_n, assuming the limits of a_n and b_n exist
  • Proposition 13.12
  • Theorem 13.16 (Monotone Convergence)
  • Theorem 14.5(c) (limit of a ratio)
  • Definition 14.12 (Cauchy sequence)
  • Proposition 14.13 (Convergent implies Cauchy)
  • Theorem 14.19 (Cauchy implies convergent)
  • Lemma 14.27
  • Explanation of what the Ultimate Sin is, and why it is a sin
  • Theorem 14.29 (Comparison test) and Corollary 14.30
• Go through your summary notes and proofs slowly, at least twice. As you do so, ask yourself questions about what the definitions and theorems mean, and how to use them. Write down your questions and your answers - you get better at writing mathematics by practicing writing mathematics!
• Re-work all homework problems. Some test questions will build directly on the homework.
• Work through the Practice Exam (ignore problems 10, 11, 12) and Solutions.