Wednesday, July 16, 2014

The radius of convergence is zero

Dear Diary,

Let's talk about a Taylor series' radius of convergence for a minute.  When you're writing down a mathematical structure for a particular object, sometimes the best you can do is generate the Taylor series in a small area of the problem.

I'll liken this to a color gradient.  If you have a panel that slowly varies from red to white like this:

If you start on the white side of the figure and move towards the red, you need to slowly add red to the spot you're standing on.  This is what a Taylor series is doing.  As you move around on the color gradient from right to left, you need to add more color...but in the physics or math problem, you add more mathematical terms to the Taylor series!

You can also a define a radius of convergence of the Taylor series.  This would mean, how far can you want with your Taylor series.  Say our gradient abruptly turned blue to the left of the above gradient.  You can't add more red to get to blue, primarily.  So, the radius of convergence of that color set up would be the width of the rectangle.

There are some things in physics that have a radius of convergence of zero.  As soon as you take one step away from where you started, the Taylor series is already bad!  Most famously, this happens in quantum field theory when you try to see how far away the electron's mass is from the bare mass in the Lagrangian.  You get an infinite distance; in other words, your Taylor series is a miserable description everywhere!  Instead, you can only use the experimental mass in your computation and then everything works!

But there are also real life examples.  For starters, consider the distribution of camping trips I've been on in my entire life.  The one I went on last week had a radius of convergence of zero about it.  You can't Taylor expand to compare it to any other camping trip!  It was that bad!  I have a bunch of trips that were good, bad, and otherwise interesting, but this one was SCARY!  I find it hard to compare to any other experience.  A description from any other experience would not be adequate.  Just like a Taylor series with a radius of convergence of zero.

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