Power Series Expansions
Up to this point we’ve been talking about power series like , where “power” refers to powers of . This led to us to show that when we evaluate a power series, the result converges in a disk centered at . But what’s so special about zero?
Indeed, we could just as well write a series like for any point . The result is just like picking up our original power series and carrying it over a bit. In particular, it still converges — and within the same radius — but now in a disk centered at .
So when we have an equation like , where the given series converges within the radius , we say that the series “represents” in the disk of convergence. Alternately, we call the series itself a “power series expansion” of about .
For example, consider the series . A simple application of the root test tells us that this series converges in the disk , of radius about the point . Some algebra shows us that if we multiply this series by we get . Thus the series is a power series expansion of about .
This new power series expansion actually subsumes the old one, since every point within of is also within of . But sometimes disks overlap only partly. Then each expansion describes the behavior of the function at values of that the other one cannot. And of course no power series expansion can describe what happens at a discontinuity.