## 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.

[...] Power Series So we know that we can have two power series expansions of the same function about different points. How are they related? An important step in this [...]

Pingback by Translating Power Series « The Unapologetic Mathematician | September 16, 2008 |

[...] Now, to be specific: if the power series converges for to a function , then has a derivative , which itself has a power series expansion [...]

Pingback by Derivatives of Power Series « The Unapologetic Mathematician | September 17, 2008 |

[...] we have power series expansions of functions around various points, and within various radii of convergence. We even have formulas [...]

Pingback by Uniqueness of Power Series Expansions « The Unapologetic Mathematician | September 18, 2008 |

[...] Okay, we know that power series define functions, and that the functions so defined have derivatives, which have power series expansions. And thus [...]

Pingback by Analytic Functions « The Unapologetic Mathematician | September 27, 2008 |

[...] what functions might we try finding a power series expansion for? Polynomials would be boring, because they already are power series that cut off after a finite [...]

Pingback by The Taylor Series of the Exponential Function « The Unapologetic Mathematician | October 7, 2008 |

[...] to stop what I was working on about linear algebra. Instead, I set off on power series and how power series expansions can be used to express analytic functions. Then I showed how power series can be used to solve [...]

Pingback by Pi: A Wrap-Up « The Unapologetic Mathematician | October 16, 2008 |