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11.1 An Introduction to Sequences & Series

11.1 An Introduction to Sequences & Series. p. 651. What is a sequence? What is the difference between finite and infinite?. Sequence :. A list of ordered numbers separated by commas. Each number in the list is called a term . For Example: Sequence 1 Sequence 2

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11.1 An Introduction to Sequences & Series

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  1. 11.1 An Introduction to Sequences & Series p. 651

  2. What is a sequence? • What is the difference between finite and infinite?

  3. Sequence: • A list of ordered numbers separated by commas. • Each number in the list is called a term. • For Example: Sequence 1Sequence 2 2,4,6,8,10 2,4,6,8,10,… Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5 Domain – relative position of each term (1,2,3,4,5) Usually begins with position 1 unless otherwise stated. Range – the actual terms of the sequence (2,4,6,8,10)

  4. Sequence 1Sequence 2 2,4,6,8,10 2,4,6,8,10,… A sequence can be finite or infinite. The sequence has a last term or final term. (such as seq. 1) The sequence continues without stopping. (such as seq. 2) Both sequences have a general rule: an = 2n where n is the term # and an is the nth term. The general rule can also be written in function notation: f(n) = 2n

  5. Write the first 6 terms of an=5-n. a1=5-1=4 a2=5-2=3 a3=5-3=2 a4=5-4=1 a5=5-5=0 a6=5-6=-1 4,3,2,1,0,-1 Write the first 6 terms of an=2n. a1=21=2 a2=22=4 a3=23=8 a4=24=16 a5=25=32 a6=26=64 2,4,8,16,32,64 Examples:

  6. The seq. can be written as: Or, an=2/(5n) The seq. can be written as: 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,… Or, an=2n+1 Examples: Write a rule for the nth term.

  7. Explicit Formula • When the rules for a sequence are written so that the nth term can be calculated immediately, then it is expressed as an explicit formula. Examples: an=2/(5n) or , an=2n+1

  8. Example: write an EXPLICIT rule for the nth term. • 2,6,12,20,… • Can be written as: 1(2), 2(3), 3(4), 4(5),… Or, an=n(n+1)

  9. Recursive Form • When given one or more of the initial terms and then defining the terms that follow using those previous terms is called a recursive formula. Example: find the 4th term if

  10. Try These … Find the sixth term of the sequences. 39 -96

  11. A Famous Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, … Found in flower petals, tree branches, bones in the human hand … THE FIBONACCI SEQUENCE

  12. Convergent – Divergent Sequences If a sequence approaches a constant as the value of n gets large, the sequence is said to converge. If a sequence does NOT converge, it is divergent.

  13. Divergent or Convergent? n 1 2 3 4 a 3 6 9 12 This sequence does not approach a constant … DIVERGENT

  14. Divergent or Convergent? Hint: for explicit formulas, graph on your calculator and evaluate what happens when x gets large. For recursive formula, find the first several terms of the sequence and determine what will happen.

  15. Divergent or Convergent? Try these …

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