Yes!!!
The coefficients are based off the fixed point of natural exponentiation! Remember the cycle with a period of about 26.9? And remember that I said that each term behaves as if it had been rotated about a quarter turn, i.e. that it was close to pi/2?
Well, take the imaginary part of the fixed point: 1.3372357...
Subtract it from pi/2 from it, and you get 0.2335606...
Now divide this into 2*pi, and guess what you get! 26.9 and change.
The coefficients of the series behave as if they are the real parts of points that are are rotating about 1.337... radians, and the root test of the first derivative indicates an exponential decline that is very close to the rate of decline of the distance of iterated logarithms from the fixed point!
So the fixed point rears its ugly head after all. Armed with this knowledge, I think we should be able to analyze the larger systems (300, 400, 500, etc.) and determine what the solution of the infinite system is going to look like.
The coefficients are based off the fixed point of natural exponentiation! Remember the cycle with a period of about 26.9? And remember that I said that each term behaves as if it had been rotated about a quarter turn, i.e. that it was close to pi/2?
Well, take the imaginary part of the fixed point: 1.3372357...
Subtract it from pi/2 from it, and you get 0.2335606...
Now divide this into 2*pi, and guess what you get! 26.9 and change.
The coefficients of the series behave as if they are the real parts of points that are are rotating about 1.337... radians, and the root test of the first derivative indicates an exponential decline that is very close to the rate of decline of the distance of iterated logarithms from the fixed point!
So the fixed point rears its ugly head after all. Armed with this knowledge, I think we should be able to analyze the larger systems (300, 400, 500, etc.) and determine what the solution of the infinite system is going to look like.
~ Jay Daniel Fox