Sect 4. Sequences

CONCEPTS

  Sequence
  Applications: summations, strings, languages
  Inductive definitions
  Principle of induction
  Recurrence

NOTATION

SKILLS
  
  1. Identify concepts in a context
     E.g., find limits of a summation
     E.g., find base case of a recurrence
     E.g., identify inductive assumption of proof by induction
  2. Manipulate summations
     E.g., prove summation theorem by induction
     E.g., find close form of a summation
  3. Understand language definition
     E.g., tell whether a string is in a language
     E.g., find product, union, closure of languages 
  4. Understand inductive definitions
     E.g., compute set inductively defined
     E.g., produce inductive definition of a set
  5. Understand induction principle
     E.g., tell structure of a proof by induction
  6. Manipulate recurrences
     E.g., find close form of a recurrence
     E.g., express recurrence for a given problem/algorithm
  
READING

  Wikibooks: Undergraduate Mathematics/Sequence
  https://en.wikibooks.org/wiki/Undergraduate_Mathematics/Sequence
  exclude "Limits and convergence" and following sections

  Wikipedia: Formal language
  https://en.wikipedia.org/wiki/Formal_language
  only paragraphs: "Words over an alphabet", "Definition",
  "Operations on languages" 

  *** Recommended ***
  Wikibooks: Calculus/Summation notation
  https://en.wikibooks.org/wiki/Calculus/Summation_notation

  Wikipedia: Summation (like the wikibook, but a bit heavier)
  https://en.wikipedia.org/wiki/Summation

  *** Recommended ***
  Wikibooks: Mathematical Proof/Methods of Proof/Proof by Induction
  https://en.wikibooks.org/wiki/Mathematical_Proof/Methods_of_Proof/Proof_by_Induction

  Youtube: Principle of Mathematical Induction Inequality Proof by The Math Sorcerer
  https://www.youtube.com/watch?v=g0EFrCYgl5s (a short example)

  Youtube: Inductive step of proving an inequality by induction by Discrete Math Zyante
  https://www.youtube.com/watch?v=GkMKPRt5jL0 (a short example)

  *** Recommended ***
  Youtube: Finding a closed form from a recursively defined sequence by Joshua Helston (short, clean example)
  https://www.youtube.com/watch?v=pHL3g3oQhyg

  *** Recommended ***   
  Lerma, sect 4.1-4.3 (exclude 4.2.3, 4.3.2)

  Smid, sect 2.2.5 (Induction)
  4.3 (Induction definions)
  4.5.2 (eample of running time)

  LLM, Sect 5 (Induction, long, extensive, entertaining)
  Chapt 21 (Recurrences, long, extensive)

  ADS, sect 3.7 (Induction)
  Chap 8 (recursion and recurrences)

  BDS, Sect 4.1-4.3 (heavy)

  Hammack, Chap 10 (!!! lots of examples and exercises !!!)

  Levin, Sect 2.5 (induction) 2.4 (recurrences)