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.
Lets look at another expansion value.
Its a little hard to see where it cuts off on the right. The following plot helps to demonstrate this. Also, the interval of convergence must be symmetrical on either side of the center of the expansion. These two factors work together to make the convergence intervals smaller as we choose expansion points closer to the vertical asymptote.