Exam 1

MCS 331 Exam 1

Part 1

Part 1 of the first exam in Real Analysis will be given in class on Monday, October 8. It will be a closed-book, closed-notes exam that tests your memory and basic understanding of the fundamental concepts and main theorems.

Bressoud: A Radical Approach to Real Analysis, chapters 1-2
- Crisis in Mathematics: Fourier Series
  - State in words what this crisis was and what it led to.
- Infinite Summations
  - What is the binomial series?
  - What is the Nested Interval(s) Principle?
  - What are Taylor series? What is Lagrange's remainder?
  - What are some problems or puzzles that arise in infinite summations?
Sprecher: Elements of Real Analysis, chapters 3 & 5 plus
- Key concepts/definitions
  - The Big Question (ps) (pdf)
  - metric, metric space
  - Cauchy sequence
  - Equivalent Cauchy sequences
  - Convergent sequence, limit of a sequence
  - The real numbers
  - Cauchy criterion
  - Archimedean property
  - Least upper bound, lub (or sup), glb (or inf)
  - Least Upper Bound Property
  - Nested Intervals Property
  - limit point
  - isolated point
  - dense in
  - closed set
  - derived set
  - closure of a set
  - bounded set
  - compact set
  - neighborhood
  - interior point
  - open set
  - interior of a set
  - connected set
- Key theorems
  - Bolzano-Weierstrass Theorem
  - Heine-Borel Theorem

Part 2

Part 2 of the first exam will be a take-home exam consisting of problems. You may use your two texts, notes, and homework in doing this exam. It will be due at the beginning of class on Wednesday, October 10. It will involve problems based on Bressoud chapters 1-2, Sprecher chapters 3 and 5, and material covered in class.

Pick up your copy of the exam outside my office, Olin 307, when you are ready to take it, and sign it out on the signout sheet. You will have about two hours to take it. Return it to my office ASAP after you finish it, and note the return time on the signout sheet.

Click here to get a postscript or pdf view of the instruction page: (ps) (pdf)