jaydfox Wrote:I'm not sure I follow, because for bases in that range, the series should be converging (e^-e < b < 1). Maybe I'm just misunderstanding what you're attempting to do?

For bases less than e^-e, the series are divergent, but the terms in the series converge exponentially on a simple alternating series, such that a summation for the divergent series should be possible with advanced methods.

Hmm, that's true... May be it is an artifact of the numerical computation of the truncated matrix. What I'm doing is, according to the rules of a geometric series for the scalar base x s(x) = 1/(1-x) and for the alternating geometric series, which occurs if x=-y this is then 1/(1+y).

To compute the above alternating sum I build

Bb = tetration-matrix/Bell-matrix to base b

Mb = (matid(n) + Bb)^-1

and then use the second column of Mb as terms for a powerseries in x

s(b,x) = sum r=0..inf mb[r,1]*x^r

This gives perfect results for the usual convergent cases (can be verified by Euler-summation of the series s(b,x) = x - b^x + b^b^x - ... + ... )

Moreover, this approach has the nice property, that using x=0 instead of x=1, I find the negative of the result already in the very first term:

This is justified by the observation that

s(b,1) = 1 - b^1 + b^b^1 - b^b^b^1 + .... = 1 - b + b^b - b^b^b + ....

s(b,0) = 0 - b^0 + b^b^0 - b^b^b^0 + .... = 0 - 1 + b - b^b - ....

and s(b,1) = - s(b,0)

But it is also

s(b,0) = sum r=0..inf mb[r,1]*0^r

s(b,0) = mb[0,1]*0^0 = mb[0,1]

and then simply

s(b,1) = - mb[0,1]

which avoids any divergent summation at all.

For some b decreasing from 0.155 to 0.15001 I get increasing values for s(b,1), say 2.0,2.1,2.2,... 2.6... , (by inspecting -mb[0,1] only) then if I approach 0.15 suddenly negative values occur, and if I decrease b to, say 1.4, I have positive values for s(b,1) again (by the according negative value in mb[0,1]). So I suspect, there is some singularity in that region, may be something like b contains reciprocal powers of e or e +- 1 .

If it is an artifact of the finite truncation this effect should change by change of dimension. I'll check & report that later...

Gottfried