Discrete Calculus & Probability

Unit 2 Outline

Summation
- Summation/sigma notation
  - Delimited form
  - General sigma notation
- Product/pi notation
- Summation methods
  - Technique 0: Look it up.
  - Technique 1: Guess the answer; prove it by induction .
  - Technique 2: Use divided differences and the interpolating polynomial
  - Technique 3: Discrete calculus
  - Technique 4: Use a computer algebra system
- Summation calculus
  - Basic manipulations
  - Integral v. summation calculus
  - Fundamental Theorem of Discrete Calculus: the telescoping sum formula
  - Antidifferences/indefinite sums (analagous to antiderivatives/indefinite integrals)
  - A short table of antidifferences
  - Converting between ordinary and factorial powers
    - Synthetic division
    - Stirling numbers of the first and second kinds
  - Summation by parts
- Solving first-order linear recurrence relations
Asymptotics
- Calculus review: behavior of functions f(x) as x -> infinity
  - Limits of rational functions at infinity
  - L'Hospital's rule
- Asymptotics definitions
  - Relations comparing functions with respect to growth rate
  - Little oh, Big Oh, Big Omega, Big Theta
- Approximating sums by integrals
- Stirling's approximation
- Exercises
  - Asymptotics exercises
  - More asymptotics exercises and answers