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 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 are 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.