We can examine this concept in further detail using a customized plot function. This special plot funcion will show where the series is within a small tolerance of e of the rigival function. This will provide us a convenient way to see where the series converges to the function and where it doesn't. It allows us to compare two approximate functions to an original function. What this diagram shows is that the series with more terms is a better fit to the function.
Lets look at another example.
We can see this even better using the special plot command. We cal also do the same thing for a different center value. Notice how the entire convergence region moves along with the center to the right. There is another interesting aspect to this investigation. The red curve is the original function.