By the way, I had another random observation about the solutions of the finite truncations of the Abel solution, (E-I)f=1. Although we're solving for the power series developed at 0, we are comparing the power series at exp(0)=1 as well.
Well, I remember when I was calculating various values of the partial solutions, I found that the values were well-behaved to about -1.4, but also to about +2.4. I loosely hypothesized that the radius of convergence of a partial truncation of, say a 400 term solution to 200 terms, would still appear to have a radius of convergence of about 1.4 centered at 0. But somehow, once all the terms of the full 400-term solution were used, there was also convergence around z=1 for a rather large radius.
Well, I think I kind of understand why, maybe. You see, if you take a truncation of a power series with a radius of convergence, and recenter the series to a point fairly distant, say more than half as far as the radius of convergence, then a good majority of the terms of the power series become garbage, and not just any garbage, but garbage with a very large root test, indicating that the radius of convergence is still limited within the original circle.
Well, what I've found is that for, say a 400-term solution to the slog base e, everything after the 150th term or so is garbage anyway, but garbage with a lower root test I might add. So it was only ever accurate to 150 terms in the first place.
After shifting the center to z=1, it's still pretty much accurate to about 150 terms, maybe just a little less, and the root test is still pretty low after the 150th term. Sure, shifting the center causes us to lose more than half our terms, but it's the half that was already garbage. And this could also be why it appears to have a decent radius of convergence at both z=0 and z=1, because it can be shifted and still have a large radius of convergence.
Well, I remember when I was calculating various values of the partial solutions, I found that the values were well-behaved to about -1.4, but also to about +2.4. I loosely hypothesized that the radius of convergence of a partial truncation of, say a 400 term solution to 200 terms, would still appear to have a radius of convergence of about 1.4 centered at 0. But somehow, once all the terms of the full 400-term solution were used, there was also convergence around z=1 for a rather large radius.
Well, I think I kind of understand why, maybe. You see, if you take a truncation of a power series with a radius of convergence, and recenter the series to a point fairly distant, say more than half as far as the radius of convergence, then a good majority of the terms of the power series become garbage, and not just any garbage, but garbage with a very large root test, indicating that the radius of convergence is still limited within the original circle.
Well, what I've found is that for, say a 400-term solution to the slog base e, everything after the 150th term or so is garbage anyway, but garbage with a lower root test I might add. So it was only ever accurate to 150 terms in the first place.
After shifting the center to z=1, it's still pretty much accurate to about 150 terms, maybe just a little less, and the root test is still pretty low after the 150th term. Sure, shifting the center causes us to lose more than half our terms, but it's the half that was already garbage. And this could also be why it appears to have a decent radius of convergence at both z=0 and z=1, because it can be shifted and still have a large radius of convergence.
~ Jay Daniel Fox
