<p>A lot of time when developing power series, many authors (including the author of that webpage) will look at particular series simply to illustrate examples of power series that have a common ratio. And in some cases, they use the knowledge of geometric series to actually create the power series in the first case.</p>
<p>For instance, if you’re looking at a geometric series, the sum for any convergent geometric series can be represented as FIRST/(1 - RATIO), where FIRST represents the first term of the series and RATIO represents the common ratio, r. This series could also be represented as FIRST + FIRST * RATIO + FIRST * RATIO ^ 2 + FIRST * RATIO ^ 3 + …</p>
<p>Well, some functions are already set up in this format. For instance, f(x) = 1 / (1 - x) is set up in that exact same format, where FIRST = 1 and RATIO = x. So connecting these ideas, we could represent 1 / (1 - x) as 1 + 1<em>x + 1</em>x^2 + 1*x^3 + …</p>
<p>And that’s really all the authors tend to do when connecting these ideas. They say, “Here’s what we already know about a geometric series that converges. Let’s construct some other series that same way.” Not all power series are constructed this way, but they use the background of geometric series as a way to introduce students to the concept of power series. The key is to recognize that not all power series are actually connected to geometric series.</p>