Power Series

We have spent quite a bit of time talking about series now and along with just only a couple of exceptions we've spent most of that time talking about how to find out if a series will converge or not.  It is now time to start looking at some particular types of series and we'll eventually reach the point in which we can talk about a couple of applications of series.

 In this part we are going to start talking regarding power series.  A power series regarding to a, or just power series, is any type of series that can be written in the form, in which a and cn are numbers.  The cn's are frequently called the coefficients of the series.  The 1^st thing to notice about a power series is that it is a function of x.  That is dissimilar from any other kind of series that we have looked at to this point.  In all the prior segments we've only allowed numbers in the series and now we are permit variables to be in the series also.  Though, this will not alter how things work.  All that we know about series still holds. 

In the conversation of power series convergence is still a main question that we'll be dealing with. The variation is that the convergence of the series will now relies on the values of x that we put into the series. A power series might converge for some values of x and not for another value of x.  Before we get too far into power series there is some terminology that we need to get out of the way.