The Unapologetic Mathematician

Mathematics for the interested outsider

Power Series Expansions

Up to this point we’ve been talking about power series like \sum\limits_{n=0}^\infty c_nz^n, where “power” refers to powers of z. This led to us to show that when we evaluate a power series, the result converges in a disk centered at {0}. But what’s so special about zero?

Indeed, we could just as well write a series like \sum\limits_{n=0}^\infty c_n(z-z_0)^n for any point z_0. 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 z_0.

So when we have an equation like f=\sum\limits_{n=0}^\infty c_n(z-z_0)^n, where the given series converges within the radius R, we say that the series “represents” f in the disk of convergence. Alternately, we call the series itself a “power series expansion” of f about z_0.

For example, consider the series \sum\limits_{n=0}^\infty\left(\frac{2}{3}\right)^{n+1}\left(z+\frac{1}{2}\right)^n. A simple application of the root test tells us that this series converges in the disk \left|z+\frac{1}{2}\right|<\frac{3}{2}, of radius \frac{3}{2} about the point z_0=-\frac{1}{2}. Some algebra shows us that if we multiply this series by 1-z=\frac{3}{2}-\left(z+\frac{1}{2}\right) we get {1}. Thus the series is a power series expansion of \frac{1}{1-z} about z_0=-\frac{1}{2}.

This new power series expansion actually subsumes the old one, since every point within {1} of {0} is also within \frac{3}{2} of -\frac{1}{2}. But sometimes disks overlap only partly. Then each expansion describes the behavior of the function at values of z that the other one cannot. And of course no power series expansion can describe what happens at a discontinuity.

September 15, 2008 Posted by | Analysis, Calculus, Power Series | 6 Comments