Refering to the header of this thread and your obvious love to iteration and trajectories, I challenge you to compute the alternating sum of all the intermediate steps of the trajectory

I, 2^I, 2^(2^I), 2^(2^(2^I)...

AS(2,I) = I - 2^I + 2^2^I - 2^2^2^I + ... -... = ??

My proposal is

AS(2,I) = -0.440033096027+0.928380628227*I

What do you think?

Gottfried

[update 4'2014] The value was computed using "regular tetration" which includes a fixpoint-shift of the exponential-series (which also leads to triangular matrices). When I wrote this posting I was not aware of the deviance of the evaluation when repelling or attracting fixpoints were chosen. A short explanation of this and a better/more meaningful estimate for this sum one can find in the mail from yesterday a couple of post below this one. Gottfried

I, 2^I, 2^(2^I), 2^(2^(2^I)...

AS(2,I) = I - 2^I + 2^2^I - 2^2^2^I + ... -... = ??

My proposal is

AS(2,I) = -0.440033096027+0.928380628227*I

What do you think?

Gottfried

[update 4'2014] The value was computed using "regular tetration" which includes a fixpoint-shift of the exponential-series (which also leads to triangular matrices). When I wrote this posting I was not aware of the deviance of the evaluation when repelling or attracting fixpoints were chosen. A short explanation of this and a better/more meaningful estimate for this sum one can find in the mail from yesterday a couple of post below this one. Gottfried

Gottfried Helms, Kassel