The Unapologetic Mathematician

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.

Like this:

Posted by John Armstrong | Analysis, Calculus, Power Series

6 Comments	»

1. [...] 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 | September 16, 2008 | Reply

2. [...] 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 | September 17, 2008 | Reply

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

Pingback by | September 18, 2008 | Reply

4. [...] 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 | September 27, 2008 | Reply

5. [...] 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 | October 7, 2008 | Reply

6. [...] 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 | October 16, 2008 | Reply

« Previous | Next »

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 24, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9397340416908264, "perplexity_flag": "head"}