For those who like pictures better than formulas, we can visualize sequences on number lines and on graphs. For those who like Kit Kats, we can visualize a giant Kit Kat bar. Either way, creating an image will help us understand better how some sequences behave.
Some sequences are well-behaved like well-trained dogs, while others are as unpredictable as wild tigers. If we plot the terms of a sequence on a number line, we can get some intuition for what the terms of the sequence are doing.
Plot the first five terms of the sequence , starting at n = 1, on a number line.
The first five terms are
Plotting these on a number line, we get
We can see that as n gets larger, the terms of the sequence are clumping around 0.
Even if we don't label the terms an on the number line, we can still tell something about what the sequence is doing.
We have all had sliced bread. It's been around since 1928. Two-dimensional graphs have been around for a while, too. While number lines are nice, we can't tell which dots go with which terms. Since we know as much about 2-D graphs as we do sliced bread, we may as well use them and see what happens.
To graph a sequence on a 2-D graph, we put n on the horizontal axis and an on the vertical axis.
For each term an of the sequence we graph the point (n, an).
Plot the first five terms of the sequence , starting at n = 1, on a graph.
We plot the points
to get this graph:
Even though the terms flip back and forth like a floundering fish, we can see that the values an get closer to 0 as n gets larger.
After going through this example, you don't need your fortune-telling turban or crystal ball to see we're heading for limits of sequences. Back the bus up for a second. First, we should go through a couple exercises to see a few more sequences plotted in two dimensions.