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Infinite Sequences

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Agenda

  • What is a sequence?
  • Infinite sequences?
  • Practice

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1. What is a sequence?

A number, then another number, and another, ...

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What is a Sequence?

  • A sequence is a list of numbers in a definite order
    • 1, 2, 3, 4, …, 10 is a sequence
    • {1, 2, 3, 4, …, 10} is a set, and not a sequence
  • For the sequence
    • There are n terms

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Special Sequences

  • Recall a few special sequences in grade 11
    • Arithmetic sequences
      • Constant difference between terms
      • ​
    • Geometric sequences
      • Constant ratio between terms
      • ​

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2. Infinite Sequences

The numbers never stop!

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Infinite Sequences

  • An infinite sequence is similar to a finite sequence
  • Except there are infinitely many terms!

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Questions About Infinite Sequences

  • When faced with an infinite sequence, all the questions about finite sequences still hold
    • What is the nth term? All the terms?
    • Is there a pattern/formula for the terms?
  • Plus some new possible questions
    • Does this tend to a specific value as the n approaches infinity?
    • Equivalent to taking the limit as the variable goes to infinity

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Convergence

  • If an infinite sequence tends to a value
    • The limit exists
    • It is called a convergent sequence
    • And it converges to the limit
  • If the infinite sequence doesn’t tend to a value
    • The limit doesn’t exist
    • It is called a divergent sequence
    • And the limit diverges

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Related to Functions

  • Taking limits to infinity for sequences is the same as finding behaviour at infinity for functions
  • For example, is there a horizontal (or oblique) asymptote?
  • What are the y-values as x approaches infinity or negative infinity?

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3. Practice

We have all the sequences!

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Practice

  • Go to the Jamboard Unit 1 Day 5 Infinite Sequences

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Credits

Special thanks to all the people who made and released these awesome resources for free:

  • Presentation template by SlidesCarnival

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