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8.1: Sequences

8.1: Sequences. A sequence is a list of numbers written in an explicit order. n th term. Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence.

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8.1: Sequences

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  1. 8.1: Sequences

  2. A sequence is a list of numbers written in an explicit order. nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences.

  3. Example: A sequence is defined explicitly if there is a formula that allows you to find individual terms independently. To find the 100th term, plug 100 in for n:

  4. Example: A sequence is defined recursively if there is a formula that relates an to previous terms. We find each term by looking at the term or terms before it: You have to keep going this way until you get the term you need.

  5. An arithmetic sequence has a common difference between terms. Example: Arithmetic sequences can be defined recursively: or explicitly:

  6. An geometric sequence has a common ratio between terms. Example: Geometric sequences can be defined recursively: or explicitly:

  7. If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term. Example:

  8. Does converge? You can determine if a sequence converges by finding the limit as n approaches infinity. (L’Hopital) The sequence converges and its limit is 2.

  9. If the sequence has a limit as n approaches infinity, it converges. • If the sequence does not have a limit, we say it diverges.

  10. If and there is an integer N for which an≤ bn ≤ cn for all n > N, then Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. Sandwich Theorem for Sequences

  11. Graph a sequence defined explicitly by an = (1/2)n • The calculator can graph sequences defined recursively or explicitly. 1. Select Seqand Dotmodes 2. Enter the expression into y = 3. Press Window and enter the following values: nMin = 1 PlotStart = 1 xmin = 0 ymin = 0 nMax = 10 PlotStep = 1 xmax = 10 ymax = 1 xscl = 1 yscl = 1 4. Press Graph

  12. Graph a sequence defined recursively by an = (1/2)an–1 a1 = (1/2) 1. Select Seqand Dotmodes • Enter the expression into y = • (the condition a1 = (1/2) goes into u(nMin) 3. Press Window and enter the following values: nMin = 1 PlotStart = 1 xmin = 0 ymin = 0 nMax = 10 PlotStep = 1 xmax = 10 ymax = 1 xscl = 1 yscl = 1 4. Press Graph

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