Taylor polynomial. System of equations for the coefficients.

[sheldonison]

This is a numerical example for base a=e

I had shown how to get 2 sides of a system of equations for solving the coefficients. One side is blue (the linear one), and the other is red (nonlinear, and I still don't know the pattern to generate the complete system of equations. I only got up to x^7, and I can get more, but of course, what I need is a more general expression.
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You might want to compare this to the intuitive slog method. Jay took that farther than anyone else, getting results accurate to 21 decimal digits using his accelerated technique. http://math.eretrandre.org/tetrationforu...php?tid=63 With 8 terms, accuracy is limited to magnitude of the eighth term, and your eighth term is actually larger than Kneser's. If you were able to generate a 20 term approximation, then theoretically, one could have a around 14 decimal digits of accuracy ... assuming the terms scale based on the nearest logarithmic singularity at sexp(-2). For base e, the Taylor's series coefficients of Kneser's solution approach log(x+2) as n gets larger.