Tommy1729, thank you for bringing this thread back to our/to my attention!

With my current knowledge I'm pretty sure, that my challenge for the alternating sum at the time of the original post was most likely false; a (still only "most likely") more meaningful version uses now the "polynomial method" for the tetration for bases outside the Shell-Thron-region, thus for instance of all real bases .

The results using this method gives the following table for some bases b and the argument x=I :

from where my challenge should have been:

I haven't (in the year of the above posts) been aware of the problem of the attracting and repelling fixpoints and the sensitivity of the "Alternating Series with increasing heights" (which I call now "alternating iteration series") around which fixpoint the series is developed. Assuming that the "polynomial method" for tetration approximates (and generalizes) the Kneser-solution and also assuming that the Kneser-solution is the best for real tetration the AS()-function should be based on that "polynomial method".

The Pari/GP-procedures are even simpler than that of the original posting;

Note, that the coefficients in C form the (head of a) powerseries, which seems to have nonzero (and surprisingly interesting) range of convergence even if base b is outside of the Shell/Thron (resp Euler)-range!

After that, do the computation:

Note finally that for bases inside the Euler-interval one can use the builtin Pari/GP-function "sumalt" to simply do a summation for the (then conditionally converging) alternating iteration series and use this to crosscheck the AS(x)-evaluation taken by the "polynomial method" for tetration.

Gottfried

With my current knowledge I'm pretty sure, that my challenge for the alternating sum at the time of the original post was most likely false; a (still only "most likely") more meaningful version uses now the "polynomial method" for the tetration for bases outside the Shell-Thron-region, thus for instance of all real bases .

The results using this method gives the following table for some bases b and the argument x=I :

Code:

`_ `

base log(base) AS(base,I)

1.1000000 0.095310180 -0.45069485+0.91385251*I

1.2000000 0.18232156 -0.40503707+0.85093158*I

1.3000000 0.26236426 -0.36504196+0.80486309*I

1.4000000 0.33647224 -0.33093412+0.77025573*I

1.5000000 0.40546511 -0.30187962+0.74327224*I

1.6000000 0.47000363 -0.27685754+0.72148441*I

1.7000000 0.53062825 -0.25501392+0.70337461*I

1.8000000 0.58778666 -0.23569667+0.68796239*I

1.9000000 0.64185389 -0.21841589+0.67458888*I

2.0000000 0.69314718 -0.20280135+0.66279493*I

2.1000000 0.74193734 -0.18856952+0.65225041*I

2.2000000 0.78845736 -0.17550012+0.64271115*I

2.3000000 0.83290912 -0.16341960+0.63399285*I

2.4000000 0.87546874 -0.15218883+0.62595235*I

2.5000000 0.91629073 -0.14169584+0.61847841*I

2.6000000 0.95551145 -0.13184894+0.61147999*I

2.7000000 0.99325177 -0.12257089+0.60488549*I

2.8000000 1.0296194 -0.11380069+0.59863552*I

2.9000000 1.0647107 -0.10548489+0.59267760*I

3.0000000 1.0986123 -0.097576064+0.58697104*I

3.1000000 1.1314021 -0.090038424+0.58148334*I

3.2000000 1.1631508 -0.082843993+0.57618212*I

3.3000000 1.1939225 -0.075964723+0.57103646*I

3.4000000 1.2237754 -0.069372706+0.56602343*I

Code:

`AS(2,I) = -0.20280135+0.66279493*I`

I haven't (in the year of the above posts) been aware of the problem of the attracting and repelling fixpoints and the sensitivity of the "Alternating Series with increasing heights" (which I call now "alternating iteration series") around which fixpoint the series is developed. Assuming that the "polynomial method" for tetration approximates (and generalizes) the Kneser-solution and also assuming that the Kneser-solution is the best for real tetration the AS()-function should be based on that "polynomial method".

The Pari/GP-procedures are even simpler than that of the original posting;

Code:

`n=32 \\ constant gives dimension for matrices`

default(realprecision,200) \\ internal real-arithmetic computation uses 200 digits

default(format,"g0.12") \\ show 12 digits in user-interface

{ASinit(b)=local(a); \\ define the matrix and coefficients for powerseries

a=log(b);

B = matrix(n,n,r,c, (a*(c-1))^(r-1)/(r-1)!); \\ the carlemanmatrix

C = matsolve(matid(n)+B,matid(n)[,2]); \\ vector of coefficients

return("ok");}

AS(x) = sum(0,n-1, x^k*C[1+k]) \\ the value for the (truncated) and approximate powerseries AS(base,x)

Note, that the coefficients in C form the (head of a) powerseries, which seems to have nonzero (and surprisingly interesting) range of convergence even if base b is outside of the Shell/Thron (resp Euler)-range!

After that, do the computation:

Code:

`base=2`

ASinit(base)

%486 = "ok"

AS( 1 ) \\ consider only x for which the truncated powerseries is convergent

%487 = 0.28740870

AS( I ) \\ reproduce the new result (if n is at least 32 )

%488 = -0.20280135 + 0.66279493*I

Note finally that for bases inside the Euler-interval one can use the builtin Pari/GP-function "sumalt" to simply do a summation for the (then conditionally converging) alternating iteration series and use this to crosscheck the AS(x)-evaluation taken by the "polynomial method" for tetration.

Gottfried

Gottfried Helms, Kassel