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Exploring Infinite Sequences in Calculus

Dive into infinite sequences, limits, arithmetic and geometric sequences in calculus. Learn to graph sequences with calculators.

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Exploring Infinite Sequences in Calculus

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  1. 9.4 Sequences

  2. What you’ll learn about • Infinite Sequences • Limits of Infinite Sequences • Arithmetic and Geometric Sequences • Sequences and Graphing Calculators … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

  3. Sequence One of the most natural ways to study patterns in mathematics is to look at an ordered progression of numbers, called a sequence. A finite sequence has a fixed number of terms. An infinite sequence has an infinite number of terms. Some sequences have a rule that gives the kth number in the sequence, others do not.

  4. Limit of a Sequence

  5. Example Finding Limits of Sequences

  6. Example Finding Limits of Sequences

  7. Arithmetic Sequence

  8. Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. –2, 1, 4, 7, …

  9. Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. –2, 1, 4, 7, …

  10. Geometric Sequence

  11. Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

  12. Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

  13. Example Constructing Sequences The third and sixth terms of a sequence are 48 and 6, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

  14. Example Constructing Sequences a3 = 48, a6 = 6. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

  15. Example Constructing Sequences a3 = 48, a6 = 6. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

  16. Example Constructing Sequences a3 = 48, a6 = 6. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric.

  17. Sequences and Graphing Calculators • One way to graph explicitly defined sequences is as scatter plots of the points of the form (k,ak). • A second way is to use the sequence mode on a graphing calculator.

  18. The Fibonacci Sequence

  19. Quick Review

  20. Quick Review Solutions

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