8.1: Sequences

nth term A sequence is a list of numbers written in an explicit order. 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.

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

A sequence is defined recursively if there is a formula that relates an to previous terms. Example: 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.

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

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

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

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

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

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 If and there is an integer N for which an ≤ bn ≤ cn for all n > N, then

The calculator can graph sequences defined recursively or explicitly. Graph a sequence defined explicitly by an = (1/2)n 1. Select Seq and Dot modes 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

Graph a sequence defined recursively by an = (1/2)an–1 a1 = (1/2) 1. Select Seq and Dot modes 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