08/24/2007, 07:07 AM
By the way, here's a comparison of the root-test for the 150-, 250-, and 400-term solutions:
The first thing to notice is that it would appear that there is a radius of convergence. After all, the root test seems to be asymptotic. The second thing to notice is that as I increased the number of terms, the values for the root test slowly climbed. So the asymptote would appear to be higher than what we see already at 400 terms. This is seen more easily in a detailed view:
But what's really weird is that, naively, the radius of convergence would appear to be at most 1/0.71 or so, about 1.4. However, the function behaves very well for real values up to about 2.4. If you view the partial sums of the series, they begin to oscillate wildly, dozens of orders of magnitude too large in absolute value. And yet, by the final term of the sequence, they settle on the correct value. Try it for yourself. Pop the last coefficient off the power series and check the radius of "good behavior", versus the original series.
I'm honestly blown away by this behavior. The series would seem to converge well outside the naive radius of convergence.
My pet theory is that the radius of convergence is around both 0 and 1, so we can go as low as -1.4 and as high as 2.4. The numbers seem to bear this out. -1.45 and 2.45 are both a few orders of magnitude too large, and -1.4 and 2.4 are both well-behaved (but starting to significant errors).
However, I'm wondering what happens if we solve even larger systems. What happens at 500, 600, 1000, 2000, 10000 terms? A million terms? Obviously there are practical limits, but can we answer these questions theoretically?
The first thing to notice is that it would appear that there is a radius of convergence. After all, the root test seems to be asymptotic. The second thing to notice is that as I increased the number of terms, the values for the root test slowly climbed. So the asymptote would appear to be higher than what we see already at 400 terms. This is seen more easily in a detailed view:
But what's really weird is that, naively, the radius of convergence would appear to be at most 1/0.71 or so, about 1.4. However, the function behaves very well for real values up to about 2.4. If you view the partial sums of the series, they begin to oscillate wildly, dozens of orders of magnitude too large in absolute value. And yet, by the final term of the sequence, they settle on the correct value. Try it for yourself. Pop the last coefficient off the power series and check the radius of "good behavior", versus the original series.
I'm honestly blown away by this behavior. The series would seem to converge well outside the naive radius of convergence.
My pet theory is that the radius of convergence is around both 0 and 1, so we can go as low as -1.4 and as high as 2.4. The numbers seem to bear this out. -1.45 and 2.45 are both a few orders of magnitude too large, and -1.4 and 2.4 are both well-behaved (but starting to significant errors).
However, I'm wondering what happens if we solve even larger systems. What happens at 500, 600, 1000, 2000, 10000 terms? A million terms? Obviously there are practical limits, but can we answer these questions theoretically?
~ Jay Daniel Fox
