| Does the sequence
4, 3, 2.5, 2.25… diverge or
converge? |
Section
1 looked at term-to-term rules. A sequence where each term is defined in
terms of a previous term is known as a recursive sequence. The terms of
a sequence are going to be refered to as u(1), u(2), u(3) and so on, where u(n)
means the nth term. Here are some examples:
| Sequence |
Recursive
definition |
| 2, 4, 6, 8... |
u(1) = 2, u(n) =
u(n-1)+2 |
| 1, 3, 5, 7… |
u(1) = 1, u(n) =
u(n-1)+2 |
| 1, 2, 4, 8… |
u(1) = 1, u(n) =
2*u(n-1) |
| 1, 3, 7, 15… |
u(1) = 1, u(n) =
2*u(n-1)+1 |
| |
|
To set up your calculator to work with
recursive sequences, select MODE and on the fourth line select
Seq. Now select Y= and you can enter the definition of a
sequence. To define the sequence 1, 3, 7, 15... you will need to set up this
display:
To display say the first 10 terms of
this sequence, select WINDOW and choose options as shown below and then
select TABLE (2nd GRAPH):
This sequence can then be displayed on
a graph. To get appropriate scales, select ZOOM... ZoomFit):
In the first graph above, the values
get bigger ansd bigger - they diverge. In the second they
converge. Plot the graphs of some more recursive sequences. Use those
given in the table above, and some of those below. What happens to the values
in these sequences? Do they diverge or converge?
| Sequence |
Recursive
definition |
| 1, 0.5, 0.25. 0.125... |
u(1) = 1, u(n) =
u(n-1)/2 |
| 4, 3, 2.5, 2.25… |
u(1) = 4, u(n) =
u(n-1)/2+1 |
| 3, 2, 1.666, 1.555… |
u(1) = 3, u(n) =
u(n-1)/3+1 |
| |
|
|
