08/28/2007, 07:46 AM
Here's the interesting part. Remember how I said I had to invert the sign of every other even term? The same thing happens for every other odd term. Moreover, the even and odd terms behave much like sine and cosine: the one near zero when the other is approaching its maximum distance from zero. Put these pieces of information together, and it appears as if we could put all four sets of numbers together, using a sine or cosine function with an argument that increases by about pi/2 for each successive term. Those who dabble in signal analysis are used to this sort of aliasing effect. After subtracting out the pi/2 per k, we're left with a little excess, which repeats about every 27 terms, perhaps closer to about 26.8 or 26.9, based on my initial studies. That's almost 28, which divided by 4 explains the nearly 7-term periodicity that Andrew observed.
When you look at only the evens or only the odds, and reverse the signs of every other term, you can make out the sine waves much more easily, and it's easy to confirm that the periodicity is less than 28 and greater than 26. More accuracy than that requires solving a larger system. My system maxes out at about 560 terms. Beyond that, 1.5 GB is no longer enough, and I start using swap space. Once maxima starts using swap space, CPU utilization drops to 20%, which means that what should take half a day to solve will take 2.5 days or more. I can't tie up my system that long, so I can only analyze the data out to 560 terms. I don't have enough data to make any solid conclusions on the solution that we're approximating with these small systems.
Anyway, moving along. I was trying to figure out why the root test initially dips, then climbs upwards towards some asymptote near 0.73. Taking the series for the logarithm as an example, I took the derivative of the power series for slog, and found the following for the root test (showing results for 200, 300, 400, 500, and 560):
Here's a detailed view:
Apologies for the axis labels, but the plotting engine in SAGE doesn't give me fine control over the axis scale and tick labels, and depending on the range I try to plot, I don't get any tick labels at all. So I have to pick very strange ranges to get the labels to appear, and I'm not entirely sure they're 100% accurate.
When you look at only the evens or only the odds, and reverse the signs of every other term, you can make out the sine waves much more easily, and it's easy to confirm that the periodicity is less than 28 and greater than 26. More accuracy than that requires solving a larger system. My system maxes out at about 560 terms. Beyond that, 1.5 GB is no longer enough, and I start using swap space. Once maxima starts using swap space, CPU utilization drops to 20%, which means that what should take half a day to solve will take 2.5 days or more. I can't tie up my system that long, so I can only analyze the data out to 560 terms. I don't have enough data to make any solid conclusions on the solution that we're approximating with these small systems.
Anyway, moving along. I was trying to figure out why the root test initially dips, then climbs upwards towards some asymptote near 0.73. Taking the series for the logarithm as an example, I took the derivative of the power series for slog, and found the following for the root test (showing results for 200, 300, 400, 500, and 560):
Here's a detailed view:
Apologies for the axis labels, but the plotting engine in SAGE doesn't give me fine control over the axis scale and tick labels, and depending on the range I try to plot, I don't get any tick labels at all. So I have to pick very strange ranges to get the labels to appear, and I'm not entirely sure they're 100% accurate.
~ Jay Daniel Fox