Computing the alternating infinite series

\( S_b(x) = x - b^x + b^{b^x} - ... + ... \)

for x=1 I came to a certain number for b, where a turning point occurs.

For b>0.15001 I have positive values for \( S_b(1) \), with decreasing b possibly increasing to e, for 0.145 < b <=0.150001... negative values, possibly diverging, and for b<0.145 again positive values. The occuring terms of the series are diverging with alternate signs in all cases, so my computations may not be well approximated. Anyway - is a number b in the near of 0.15 known as significant in any other context in tetration?

\( S_b(x) = x - b^x + b^{b^x} - ... + ... \)

for x=1 I came to a certain number for b, where a turning point occurs.

For b>0.15001 I have positive values for \( S_b(1) \), with decreasing b possibly increasing to e, for 0.145 < b <=0.150001... negative values, possibly diverging, and for b<0.145 again positive values. The occuring terms of the series are diverging with alternate signs in all cases, so my computations may not be well approximated. Anyway - is a number b in the near of 0.15 known as significant in any other context in tetration?

Gottfried Helms, Kassel