It quite looks as if all coefficients are positive for \( 1<b\le\eta \) (and mixed for \( b>\eta \)). In this case one can sureley use the strict increase of the coefficients plus bound to show that the coefficients converge.
As far as I know there was nothing published yet about the convergence of the series (which as wrote elsewhere was already considered by Peter Walker and he stated there that the problem of convergence is open).
As far as I know there was nothing published yet about the convergence of the series (which as wrote elsewhere was already considered by Peter Walker and he stated there that the problem of convergence is open).
