Computing Andrew's slog solution

[jaydfox]

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).

Actually, this makes sense. The equation we solved was at two locations, not at one as with a traditional power series. In other words, the system of equations we solved was set up to ensure that we got exact values at x=0 and x=1, not only for the y value, but for the first k derivatives as well. We could do a polynomial shift to center the power series at x=1, and we'd probably observe a similar root test. (Note to self: test this). Therefore, there could be a radius around both points. I'm not sure if this would imply a figure-8 shape, or an ellipse with two foci. Worth testing. (The 150-term graph in Andrew's paper on page 18 seems to imply a figure-8 type of shape, though the image didn't compress well, so I can't make out any labels for reference.)

However, I'm basing all this off observations made from the 400-term series. Perhaps it's just a coincidence. More testing needed...